Journal of
Low Power Electronics
and Applications
Tutorial
Graph Coloring via Locally-Active Memristor
Oscillatory Networks
Alon Ascoli 1, * , Martin Weiher 1 , Melanie Herzig 2 , Stefan Slesazeck 2 , Thomas Mikolajick 2,3
and Ronald Tetzlaff 1
1
2
3
*
Citation: Ascoli, A.; Weiher, M.;
Herzig, M.; Slesazeck, S.; Mikolajick,
T.; Tetzlaff, R. Graph Coloring via
Locally-Active Memristor Oscillatory
Networks. J. Low Power Electron. Appl.
2022, 12, 22. https://doi.org/
10.3390/jlpea12020022
Academic Editors: Alexander Serb
and Adnan Mehonic
Received: 22 January 2022
Accepted: 2 April 2022
Published: 18 April 2022
Publisher’s Note: MDPI stays neutral
with regard to jurisdictional claims in
Chair of Fundamentals of Electrical Engineering, Institute of Circuits and Systems, Faculty of Electrical and
Computer Engineering, Technische Universität Dresden, 01062 Dresden, Germany;
martin.weiher@tu-dresden.de (M.W.); ronald.tetzlaff@tu-dresden.de (R.T.)
Nano-Electronic Materials Laboratory (NaMLab) gGmbH, 01187 Dresden, Germany;
melanie.herzig@namlab.com (M.H.); stefan.slesazeck@namlab.com (S.S.);
thomas.mikolajick@namlab.com (T.M.)
Institute für Halbleiter-und Mikrosystemtechnik, Technische Universität Dresden, 01062 Dresden, Germany
Correspondence: alon.ascoli@tu-dresden.de
Abstract: This manuscript provides a comprehensive tutorial on the operating principles of a bioinspired Cellular Nonlinear Network, leveraging the local activity of NbOx memristors to apply
a spike-based computing paradigm, which is expected to deliver such a separation between the
steady-state phases of its capacitively-coupled oscillators, relative to a reference cell, as to unveal
the classification of the nodes of the associated graphs into the least number of groups, according
to the rules of a non-deterministic polynomial-hard combinatorial optimization problem, known as
vertex coloring. Besides providing the theoretical foundations of the bio-inspired signal-processing
paradigm, implemented by the proposed Memristor Oscillatory Network, and presenting pedagogical examples, illustrating how the phase dynamics of the memristive computing engine enables
to solve the graph coloring problem, the paper further presents strategies to compensate for an
imbalance in the number of couplings per oscillator, to counteract the intrinsic variability observed
in the electrical behaviours of memristor samples from the same batch, and to prevent the impasse
appearing when the array attains a steady-state corresponding to a local minimum of the optimization
goal. The proposed Memristor Cellular Nonlinear Network, endowed with ad hoc circuitry for the
implementation of these control strategies, is found to classify the vertices of a wide set of graphs
in a number of color groups lower than the cardinality of the set of colors identified by traditional
either software or hardware competitor systems. Given that, under nominal operating conditions, a
biological system, such as the brain, is naturally capable to optimise energy consumption in problemsolving activities, the capability of locally-active memristor nanotechnologies to enable the circuit
implementation of bio-inspired signal processing paradigms is expected to pave the way toward
electronics with higher time and energy efficiency than state-of-the-art purely-CMOS hardware.
Keywords: graph coloring; cellular nonlinear networks; memristor oscillatory networks; locally-active
memristors; control theory
published maps and institutional affiliations.
1. Introduction
Copyright: © 2022 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
Memristor technologies promise to revolutionise the world of electronics in the years
to come, allowing to boost the performance of integrated circuits beyond the Moore era.
Theoretically introduced in 1971 by L. Chua [1], memristors are essentially resistances
with state- and input-dependent programmable capability [2]. Despite the first association
between Chua’s theory and the experimental observation of fingerprints of memristive behaviour at the nanoscale was made by R.S. Williams and his team at Hewlett Packard Labs
in 2008 [3], the appearance of memory resistance switching effects in miniaturized physical
structures were reported in numerous occasions throughout the past two centuries [4],
J. Low Power Electron. Appl. 2022, 12, 22. https://doi.org/10.3390/jlpea12020022
https://www.mdpi.com/journal/jlpea
J. Low Power Electron. Appl. 2022, 12, 22
2 of 30
constituting the object of extensive and intensive investigations for the development of
novel solid-state memories first in the 1960s [5]. In fact, the main application of memristors [6]—certainly the most profitable one from a business perspective point of view—is the
design of memories with higher retention, lower power consumption, and larger density
as compared to state-of-the-art data storage units [7]. Another major field of application
regards the development of innovative neuromorphic systems, which resemble biological
entities more closely than traditional artificial networks ([8–11]). A closely-related branch
of research takes inspiration from the high levels of organization and energy efficiency
of biological systems to develop mem-computing machines ([12–14]), which extend the
functionalities of traditional cellular architectures [15]. Another bio-inspired research direction aims to the circuit implementation of in-memory computing paradigms through
high-capacity memristive memories, stacked in 3D crossbar arrangement above underlying
CMOS circuitry [16], and employed, alternatively, to store data or to execute processing
tasks ([17–20]), which reveals the high potential of novel hardware platforms of this kind to
resolve the von Neumann bottleneck, limiting the maximum operating speed of traditional
computing engines, in the near future. Additionally, the unique combined capability of
non-volatile resistance switching memories to sense data ([21,22]), learn how to recognize
patterns [23], process information [24], and store multiple states [25] within a single tiny
physical volume, and the availability of nanoscale locally-active [26] volatile memristors,
which may amplify the small signal upon suitable polarization ([27,28]), open up yetunexplored opportunities for the Internet-of-Things (IoT) industry, which urgently calls for
the development of miniaturized, low-power, light-weight, portable, and smart technical
systems, which, within a very short time frame, are able to acquire a large amount of
information from the environment, to extract features of interest from noisy data so as to
solve specific optimization problems, and to store or transmit to a prescribed user the most
relevant results of the computation.
One of the most challenging tasks for computing machines, based upon the classical
von Neumann architecture, is the solution of combinatorial optimisation problems belonging to the non-deterministic polynomial (NP)-hard complexity class1 . Nowadays there is a
huge interest in developing novel inexpensive low-power high-speed hardware solutions
capable to solve NP-hard problems more efficiently than traditional computers, given that
high-performance technical systems of this kind may find application in various industry
sectors, e.g., for traffic management, airline scheduling, gene sequencing, and electronic
chip wiring.
A recent Nature Electronics publication [29] proposed the use of a memristive crossbar
array—refer to Figure 1—for accelerating the vector-matrix multiplications (VMMs) at
the basis of the update rule of an iterative machine learning algorithm, which is credited
to Hopfield, and enables to solve combinatorial problems, within an overall disruptive
analogue hardware architecture, leveraging and controlling its numerous inherent noise
sources to allow a power-efficient derivation of the optimal solutions.
1
The time it takes for a von Neumann computing machine to find the optimal solution to a NP-hard problem,
which involves n elements, scales exponentially with n.
J. Low Power Electron. Appl. 2022, 12, 22
3 of 30
Figure 1. In-memory computing in a N × N memristive crossbar array (here N = 4). In-memory
computing in an N × N memristive crossbar array (here, N = 4). Naturally obeying Kirchhoff’s
Current Law (KCL), the bio-inspired network enables a time-efficient computation of VMMs. With j ∈
{1, . . . , N }, the current flowing down the jth column of the array is simply given by i j = ∑iN=1 Gi,j · vi ,
where Gi,j denotes the conductance of the memory resistive switch located at the intersection between
the conductive nanowires stretching along row i and column j [29]. The computation of the currents
at the outputs of the crossbar columns assumes that the bottom terminals of all the memristors—refer
to the thick black horizontal segments in their circuit-theoretic symbols—are at virtual ground.
Cellular Nonlinear Networks (CNNs) with locally-active volatile memristors [30]
constitute a powerful engine for the implementation of bio-inspired spike-based computing
paradigms. This manuscript, inspired to a recent publication [31], is devoted to explain in
a pedagogical form how these bio-inspired memristor cellular arrays2 , may be adopted
for solving a complex NP-hard problem known as vertex coloring. While page limitation
prevented a complete description of the theory at the basis of the spike-based computing
paradigm implemented by the proposed cellular arrays3 , all details are pedagogically
reported in this tutorial. In regard to the structure of the manuscript, Section 2 provides a
brief description of the memristor model adopted for the study. Section 3 introduces the
memristive computing engine for solving the vertex coloring problem, including a discussion of its operating principles, and the specification of strategies aimed to compensate for
non-idealities, including an imbalance in the number of couplings per oscillator, and the
memristor device-to-device variability. Section 4 presents a rigorous iterative procedure
for coloring a graph via the network phase dynamics. Importantly, control paradigms to
resolve local minima-based impasse conditions [33] are proposed in Section 5, which further
compares the performance of the proposed spike-based computing engine, endowed with
ad hoc circuitry to implement such strategies, with the solutions of state-of-the-art software
and hardware competitor systems. Finally, conclusions are drawn in Section 6.
2
3
CNNs with non-volatile memristors ([12–14]) may pave the way toward the development of advanced
visual-sensor processors [32] with high spatial resolution and intrinsic memory capability.
Importantly, depending upon the graph under focus, the proposed Memristor CNN (M-CNN) may feature
either local or non-local capacitive couplings. The solution of the vertex coloring problem through the proposed
M-CNNs depends upon the phase differences among the oscillations developing in the constitutive units of the
array at steady state. In order to highlight the steady-state oscillatory behaviours of the cells during operation,
the bio-inspired arrays are also referred to as Memristor Oscillatory Networks (MONs) in the remainder of
the manuscript.
J. Low Power Electron. Appl. 2022, 12, 22
4 of 30
2. A Physics-Based Model for the Threshold Switching Dynamics of a Nano-Scale
Locally-Active Memristor Device Stack
In a past study [34,35] we employed physics laws to construct state evolution and
memductance functions of a NbOx memristor, fabricated at NaMLab, after inferring the
physical mechanisms, which underlie its nonlinear dynamics, from the outcome of experimental measurements, and the insights gained through theoretic investigations of a
mathematical model, derived previously on the basis of Chua’s Unfolding Theorem [28]. A
thorough analysis of the proposed physics-based model [34] revealed that the Mott insulatorto-metal transition does not constitute the key physical mechanism at the origin of the threshold
switching process, that each of our NbOx -based memristors undergoes under the application of
a generic quasi-static voltage stimulus between its two terminals. Conversely, a temperatureactivated trap-assisted Poole-Frenkel conduction mechanism underlies the abrupt turn-on
dynamics of the volatile memristor. Importantly, shortly after our discovery, a further proof
of evidence for the validity of our conjecture was provided by engineers from Hewlett
Packard (HP) Labs [36]. Remarkably, despite it was originally proposed for the NbO microstructure, the physics-based model in [34] was found to fit rather well also experimental
data extracted from nanoscale variants of the NbOx threshold switching resistance from
NaMLab after minor adaptations [37,38].
As illustrated in Figure 2(a), ([30,31]), the equivalent circuit model of the memristor
M consists of the series combination between a linear resistor Rc , capturing the action of
the top electrode resistance, and a parallel one-port, formed by a core memristor M̃, and a
nonlinear resistor R, which accounts for the parasitics inherent to the NbOx nanostructure,
and is responsible for the manifestation of leakage current effects. Several studies [34,35]
have revealed that the state of the core memristor is well captured by its body temperature T.
In fact, as anticipated earlier, threshold switching effects in the nano-device originate from
runaway Joule self-heating governed by Poole-Frenkel electrical conduction mechanisms.
Taking this into account for the formulation of a state-dependent Ohm’s law, with ṽm (ĩm )
representing the voltage (current) of M̃, and choosing Newton’s law of cooling to dictate
the time evolution of the state, the DAE set, governing the static and dynamic behaviour of
the core memristor may be expressed as
dT
dt
=
ĩm
=
1
Γ
· ĩm · ṽm − th · ( T − Tamb ),
Cth
C
th
1
a01 − a11 · |ṽm |
· ṽm ,
G ( T, ṽm ) · ṽm ,
· exp −
R01
T
g( T, ṽm ) ,
(1)
(2)
where Cth (Γth ) stands for the effective thermal capacitance (conductance) of the core device
M̃, Tamb denotes the ambient temperature, R01 , a01 and a11 are constants4 , while vm (im )
symbolises the voltage (current) falling across (flowing though) the memristor M, whose
circuit-theoretic symbol is shown in Figure 2(b). Remarkably, the DAE set (1)–(2) of the
core memristor falls into the voltage-controlled extended memristor family from Chua’s
classification5 [40]. The constitutive relationship f (vR , iR ) = 0 of the nonlinear resistor R
is given in implicit form as
!
p
a02 − a12 · |vR |
vR = R02 · iR · exp
.
(3)
Tamb
4
5
The reader is invited to consult [34,35] for details on the association between the real parameters R01 , a01 and
a11 and the physical properties of the core nanostructure.
It is instructive to observe that there exist memristor physical realisations, whose models feature an even more
general input- and state- dependent Ohm law than what is admissible for extended memristors. For these
two-terminal devices, including the TiO2 memristor from HP Labs [39], the mathematical description includes
an implicit Ohm law of the form h( x, vm , im ) = 0, with x, vm , and im denoting the device state, voltage, and
current, respectively.
J. Low Power Electron. Appl. 2022, 12, 22
5 of 30
revealing that a variant of the Poole-Frenkel law explains current transport phenomena in
the parasitic resistor as well6 . The nominal parameter values for the core memristor and
nonlinear resistor models were obtained by fitting the underlying equations to experimental
data extracted from nano-device samples (see [37,38] for details on the NbOx nanostructure
fabrication process). Importantly, since the non-negligible intrinsic spread in dynamic
behaviour from sample to sample may impair the capability of a computing engine based
on memristive hardware to perform a predefined data processing task as desired, this nonideality may not be neglected in circuit design considerations. Particularly, in the analysis
to follow, where a Memristor Oscillatory Network (MON) is adopted to find optimal
solutions to graph coloring problems [31], it will be accounted through the replacement of
specific parameters in the NbOx nano-scale threshold switch physics model, specifically
Γth , R01 , a01 , a11 , RC , R02 , and a12 , with corresponding ones, i.e., in turn, Γth,α , R01,α , a01,α ,
a11,α , RC,α , R02,α , and a12,α , that are controlled via a real variable α, which is set randomly
to a distinct value chosen from a uniform distribution across the closed range [0, 1] for
each nanostructure individually, prior that the simulation of the ODE, modelling the array
associated to a pre-specified graph, is commenced.
Figure 2. (a) Equivalent circuit of the physical model of a NbOx nanoscale memristor M from
NaMLab. The linear resistor RC and the nonlinear resistor R respectively account for the effects of
electrode contact resistance and parasitics. (b) Memristor circuit-theoretic symbol.
Table 1 reports the parameter setting of the nano-scale memristor physics-based model
modulated according to the device-to-device variability estimated statistically beforehand
through the analysis of current-voltage characteristics of a large number of samples under
a common quasi-static stimulation [31].
Table 1. Parameter setting for NbOx nanoscale threshold switch from NaMLab. The effects of
the memristor-to-memristor variability are accounted through the assignment of a distinct value,
chosen randomly within the closed set [0, 1] to the variable α, controlling specific coefficients of
Equations (1), (2), and (3).
Cth / J · K−1
1 · 10−14
Γth,α / W · K−1
1.889 · 10−6 · 1.064α
Tamb / K
293
R01,α / Ω
3.047 · 0.831α
a01,α / K
3620 · 1.061α
a11,α / K· V−1
820.4 · 1.137α
Rc,α / Ω
173.8 · 1.092α
R02,α / Ω
565 · 1.377α
a02 / K
1000
a12,α / K· V−1/2
168.8 · 1.083α
3. Memristive Computing Engine for Solving the Vertex Coloring Problem
One of the most popular NP-hard problems is graph or vertex coloring. Given an
undirected graph, consisting of a certain arrangement of edge-coupled vertices, the aim
of the problem is to assign a color to each vertex, satisfying the constraint, which dictates
6
The mathematical description of the nonlinear resistor, formulated in Equation (3), is in fact equivalent to the
model of a NbOx memristor, as originally presented in [34,35], which reveals the correspondence of the real
parameters R02 , a02 and a12 to physical properties of the nanostructure, in the limit when changes occurring in
the device state, defined as its body temperature, are negligible.
J. Low Power Electron. Appl. 2022, 12, 22
6 of 30
that a given color should be shared between as many vertices as possible so long as no
edge connects any two of them. The lowest number of color groups, the vertices of the
unconnected graph may be classified into, is called chromatic number.
As revealed back in 1988, graph coloring may be achieved by harnessing synchronisation mechanisms in arrays of of coupled oscillatory cells [41]. A more recent work [42]
showed that the assignment of colors to vertices of a graph may be naturally implemented
through the analysis of the steady-state phase shifts between capacitively-coupled relaxation oscillators exploiting negative differential resistance (NDR) effects in vanadium
dioxide (VO2 ) nano-structures. Inspired from this research, we have recently investigated
the capability of an oscillatory network, leveraging locally-active dynamics in NbOx memristors, and featuring capacitive couplings, to identify the minimum possible number of
colors assignable to the vertices of an associated undirected graph via phase dynamics.
In order to color a given graph of N vertices or nodes, a unique number in the set
{0, 1, . . . , N − 1} is first attributed to each vertex. A one-to-one association is then established between the vertices (edges) of the graph, and the oscillators (coupling capacitors,
each of capacitance CC ) of the associated network. The oscillator 0, corresponding to the
node 0, assumes a critical role in the solution of the graph coloring problem, and is called
reference cell. A general indication, regarding the selection of a suitable reference oscillator
among the N possible candidates, will be given shortly. As an example, Figure 3(a) and (b)
show a 6-node ring and the associated MON, respectively.
Figure 3. (a) A 6-node ring-shaped undirected graph (b) Associated MON. The oscillator i of the
network corresponds to the vertex i of the graph (i ∈ {0, 1, 2, 3, 4, 5}).
With reference to Figure 4, plots (a) and (b) show the oscillator circuit and its symbol,
respectively. Each oscillatory cell is composed of the parallel connection between a NbOx
memristor M, a bias circuit, consisting of the series combination of a DC voltage source VS
with a series resistor RS , allowing to polarize [28] the resistance switching memory within
the locally-active region of its DC current-voltage locus, and a capacitor C.
Figure 4. Memristive oscillatory cell (a) and its symbol (b).
On the basis of the NbOx nano-device physics-based model, expressed by Equations (1),
(2), and (3), under the variability-aware parameter setting of Table 1, we carried out a deep
numerical investigation of the capability of an array of capacitively-coupled memristive
J. Low Power Electron. Appl. 2022, 12, 22
7 of 30
oscillatory cells to classify the vertices of a pre-defined undirected graph in the least number
of groups.
While the memristor model parameters, accounting for the inherent fluctuations in
static and dynamic properties among distinct nano-device samples, were already reported
in Table 1, only the values assigned to the physical quantities of the non-memristive circuit
elements in the proposed MON are provided in Table 2.
Table 2. Parameter setting for the non-memristive circuit elements in the oscillator of Figure 4(a).
VS / V
2.5
RS / Ω
C/ F
CC / F
5525
10 · 10−9
0.2 · 10−9
3.1. Operating Principles of the Capacitively-Coupled Networks
The capacitive nature of the couplings in the network is responsible for pulling the
phases of physically-connected oscillators far apart one from the other at steady-state. This
repelling mechanism may be exploited to colour the vertices of the associated graph. The
phases of uncoupled oscillators tend to form clusters, which may be interpreted as color
groups for the corresponding vertices. The larger is the separation between phase clusters,
the simpler is the classification of the nodes of the graph into color groups.
To illustrate this concept, let us consider a simple undirected graph, composed of
one edge, which couples two nodes, as shown in Figure 5(a). The chromatic number of
this graph is obviously equal to 2. The corresponding oscillatory network is depicted in
plot (b) of the same figure. Simulating the circuit with nominal parameter setting7 , the
time waveforms of the voltages across the capacitors or those of the currents through
the memristors within the circuits of the two capacitively-coupled memristive oscillators
are expected to feature a steady-state phase shift of about 180◦ . Upon the emergence of
anti-phase synchronisation between the two oscillators of this simple network, it would be
natural to assign one color to vertex 0 and another one to vertex 1 of the associated graph
of Figure 5(a).
As a further example, Figure 5(c) visualises another undirected graph with chromatic
number equal to 2. The respective oscillatory array is shown in plot (d) of the same figure.
Simulating this network, the phases of oscillators 1 and 2 are expected to cluster together,
and to shift away from the phase of reference oscillator 0 as much as possible, approaching
a relative value of approximately 180◦ at steady state. With the network exhibiting such
a phase pattern at steady state, it would be straightforward to divide the vertices of the
associated graph into two color groups, including vertex 0 and vertices 1 and 2, respectively.
Due to a couple of non-idealities the expectations on the phase dynamics of the
networks in plots (b) and (d) of Figure 5 are not fulfilled in practice. Details will be
provided in the next two sections.
7
The nominal parameter setting is obtained from Table 1 for α = 0.5. As will be shown later, simulating the
memristor model under a quasi-DC voltage stimulus and with the variability parameter stepped across its
existence domain, the locus observed for α = 0.5 appears in the center of the distribution of characteristics
emerging in the voltage-current plane.
J. Low Power Electron. Appl. 2022, 12, 22
8 of 30
Figure 5. (a) A 2-node 1-edge graph. Its chromatic number is 2. (b) Oscillatory network corresponding
to the graph in (a). (c) A 3-vertex 2-edge graph. Its chromatic number is once again 2. Interestingly,
the number of edges departing from vertex 0 (from either vertex 1 or vertex 2) is 2 (1). (d) Oscillatory
network corresponding to the graph in (c).
Before proceeding, the following remark explains how the autonomous memristive
array is initialised, and clarifies how the steady-state phases of the cells are computed when
the network converges to an oscillatory solution.
Remark 1. With VS and RS fixed to specific values, as reported in Table 1, all memristors in the
network are biased in a common operating point lying along the NDR of the DC Im –Vm locus
of the NbOx resistive nanoswitch. Defining as vC,i and Ti the states of the second-order cell i of
the memristive array (i ∈ {0, . . . , N − 1}), the initial conditions for the capacitor voltage and
the memristor8 temperature are set in each oscillator to 0 V, and to the ambient temperature Tamb ,
fixed to 293 K in Table 1, respectively. A random sequence is generated to mismatch, in a nondeterministic way, the time instants, at which the signals generated by the DC voltage sources
within the oscillators ramp toward the nominal VS value over a time span of 1 µs at the beginning
of a simulation. If sustained periodic oscillations develop across the network at steady state, for
each oscillator i ∈ {0, . . . , N − 1}, the phase of the memristor current im,i relative to the phase of
the current im,0 through the NbOx device in the reference oscillator 0 is then computed, as follows.
First, the common period T of the oscillations, observed in the time waveforms of the memristor
currents at steady state, is estimated. For each i-value in the set {0, . . . , N − 1}, the time instant
ti , at which the memristor current im,i in the cell i attains a given threshold value Ith , specifically
0.5 mA, during its ascending phase, within a single common steady-state cycle, is then recorded.
The cycle, utilised for these calculations, covers the time span [t0 , t0 + T ], where t0 marks the time
instant, when this threshold crossing event occurs for the memristor current im,0 in the reference cell
0. Next, for each i-value, the temporal span ∆ti , ti − t0 , which separates the instants ti and t0 , at
which the threshold crossing event occurs for the memristor currents in the cells i and 0, respectively,
is calculated. Finally, this allows to compute the steady-state phase shift between cells i and 0 via9
(s)
ϕi
, ω0 · ∆ti , where ω0 ,
2π
T ,
for each i-value.
3.2. Compensation for an Imbalance in the Number of Couplings per Oscillator
If an imbalance in the number of edges per node characterises the coupling structure of
a given graph, the phases of physically-coupled oscillators in the corresponding memristive
oscillatory network may be found to hold only a marginal distance one from the other at
steady state. The balanced nature of the graph of Figure 3 (Figure 5(a)) originates from
the fact that each of its six (two) nodes is coupled to 2 other nodes (the other node). On
the other hand, inspecting the graph of Figure 5(c), vertex 0 is coupled to vertices 1 and 2,
while each of vertices 1 and 2 is connected to vertex 0 only. The resulting imbalance in the
number of couplings per cell, arising in the associated array of memristive oscillators—refer
to Figure 5(d)—prevents the phases of cells 1 and 2 from separating as expected from the
8
9
The voltage across (current through) the memristor in the cell i is indicated via vm,i (im,i ).
(s)
The units of the steady-state relative phase of oscillator i, computed via ϕi = ω0 · ∆ti , are radiants. In order
(s)
to express ϕi
in degrees, its formula needs to be scaled by the factor
180◦
π
(i ∈ {0, . . . , N − 1}).
J. Low Power Electron. Appl. 2022, 12, 22
9 of 30
phase of cell 0. This is shown in the phase diagram10 of Figure 6(a), where the dashed orange
and green traces, illustrating the time evolution of the phases of cells 1 and 2 relative to cell 0,
respectively, feature a separation as small as 50◦ from the reference 0◦ level at steady state. The
unbalanced nature of the network of Figure 5(d) results in an imbalance in the capacitive load
per oscillator. Looking at the coupling configuration in this array, and recalling the circuit of
each oscillator, shown in Figure 4(a), in which, for simplicity, the memristor is replaced with its
small-signal equivalent circuit model [43], the application of basic circuit-theoretic principles
allows to obtain an expression in the sinusoidal regime for the capacitive impedance ZCi ( jω),
loading oscillator i for each value of i in the set {0, 1, 2}, i.e.,
= ZC k ( ZCC ( jω ) + ZC ( jω )) k ( ZCC ( jω ) + ZC ( jω )),
= ZC k ZCC ( jω ) + ZC ( jω ) k ZCC ( jω ) + ZC ( jω ) ,
ZC2 ( jω ) = ZC k ZCC ( jω ) + ZC ( jω ) k ZCC ( jω ) + ZC ( jω )
,
ZC0 ( jω )
(4)
ZC1 ( jω )
(5)
(6)
where
ZC ( jω )
=
ZCC ( jω )
=
1
, and
jωC
1
.
jωCC
(7)
(8)
Defining
Ca k Cb
,
Ca · Cb
,
Ca + Cb
(9)
the load capacitance Ci of the oscillator i ∈ {0, 1, 2} in the network of Figure 5(d) may be
extracted easily from the (i + 1)th equation in the triplet (4)–(6), yielding
C0
C1
C2
= C + 2 · (CC k C ),
= C + CC k (C + CC k C ) ≈ C + CC k C,
= C + CC k (C + CC k C ) ≈ C + CC k C,
(10)
(11)
(12)
where the approximations stem from the inequality C >> CC , yielding C + CC k C ≈ C.
Analysing Equations (10)–(12) it is clear that, in order to compensate for the imbalance in
the capacitive load per oscillator within the network of Figure 5(d), an additional capacitor
with capacitance Ccomp,j = CC k C should be added in parallel to the capacitor C in the
circuit of oscillator j, for each j-value in the set {1, 2}, as shown in Figure 6(b). Simulating
the balanced network, the relative phases of cells 1 and 2 cluster together, converging to
values close to the expected 180◦ level at steady state, as may be evinced from plot (a) in
the same figure, where the orange and green solid traces reveal the time evolution of ϕ1
and ϕ2 , respectively.
10
A new graphical tool—which we call phase diagram—is introduced in this research study [31] for visualising
the phase dynamics of the network. Referring, for example, to the phase diagram of Figure 6(a), a specific trace
visualises the time evolution of the phase of oscillator j relative to oscillator 0 (j ∈ {1, . . . , N − 1}). Reading the
time flow along the radial direction, the angle between the segment, joining the origin to the point, where the
trace is found to lie at time t, and the blue horizontal line, denoting the 0◦ -valued reference level, represents
the phase shift ϕ j (t) of oscillator j with respect to oscillator 0 at time t.
J. Low Power Electron. Appl. 2022, 12, 22
10 of 30
ONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS
3
(a)
90°
12
45°
135°
α=0.0
α=0.2
α=0.4
α=0.6
α=0.8
α=1.0
3ms
2ms
1ms
1
180°
2
0ms
225°
0° 0
315°
270°
0.2
0.4
0.6
vM /mA
0.8
1.0
c current-voltage characteristic of distinct realizations of
depending on the variability parameter α, which accurately
bution obtained by measuring 196 NbOx-based threshold
d number of connections
Fig. 1(a) will be implemented with its associn Fig. 1(b), we can determine the evolution of
between the oscillators, as depicted in Fig. 4
es. For this phase shift oscillator 0 will serve
nd a large phase shift relative to this reference
eted as a different color. On the radial axis, the
and the angle in this plot illustrates the phase
numbers represent the number of the respective
ad coupled only two oscillators by a capacitor,
arameters of Table I, we would have recorded a
approximately 180° between them, as known
[9], [15]–[18]. Thus, similarly, it would be
hieve a phase shift of 180° between oscillators
ell as between oscillators 0 and 2, to obtain
maximum phase separation between different
ndently of the chosen graph. But, as seen in
lines) we derive approximately 50° phase shift.
e because oscillator 0 has two connections and
nd 2 only have one. Because of this, there is
i
n the network. The effective capacitance Cef
f
cillator i ∈ {0, 1, 2}, consists of the internal
of the oscillator and in parallel to this, for each
ly-connected neighboring oscillators, the series
ween the respective coupling capacitor Cc and
itance loading the respective coupling capacitor
us, we derive the following equations for the
s of this network:
+ 2 · (CC ||C)
(3a)
+ CC || (C + CC ||C)
≈
C + CC ||C
(3b)
+ CC || (C + CC ||C)
≈
C + CC ||C
(3c)
for oscillator 1, which has only one physically-connected
ator, i.e. the reference oscillator 0, the total capacitance
tive coupling capacitor is the parallel between the internal
reference oscillator 0 and the series connection between the
h couples oscillators 0 and 2, and the internal capacitance
Fig. 4: Evolution of the phase shift (referenced to oscillator 0) of the three
oscillators for the graph of Fig. 1 for the unbalanced (dashed) and balanced
(solid) case without variability (α = 0) between the memristors. The bold
numbers represent the numbering of the respective vertices.
Figure 6. (a) Phase diagram visualising the time evolution of the phases of oscillators 1 (in orange)
and 2 (in green) relative to oscillator 0, sitting on the 0◦ phase state throughout the simulation (blue
horizontalCCline)
CC for the original unbalanced network of Figure 5(c) (see the dashed traces) and for
the compensated network in plot (b) of this figure (refer to the solid traces). In the first (latter) case,
CC ||C
CC ||C
1 of 0oscillators
2
the phases
1 and 2 are found to cluster together, and to distance themselves from the
◦
reference 0 phase, associated to oscillator 0, by approximately 50◦ (180◦ ) at steady state. (b) Complete
Fig. 5: Balanced associated network of the graph depicted in Fig. 1.
circuitry of the memristive oscillatory network of Figure 5(d) after compensation for the imbalance in
thewith:
number of couplings per oscillator. Here Ccomp,1 = Ccomp,2 = CC k C.
since CC C,
CC ||C =
CC C
.
CC + C
With reference
to Figure 6(b), it is interesting to observe that the compensating caThe difference in this effective capacitance is approximately
pacitance
C
for
oscillator j ∈ {1, 2} is equal to the product between CC k C and
comp,j
the difference in the number of connections times CC ||C. If we
add this difference
the oscillator ofbetween
the associated the
network
the todifference
number of connections for oscillator 0, coinciding with the
(as illustrated in Fig. 5), the network becomes balanced.
number
of connections
per oscillator in the unbalanced network of Figure 5(d),
The evolutionmaximum
of the phase shift
for the balanced
associated
network is depicted
in Fig.number
4 (solid lines).
As
expected, the
and
the
of
connections
for
oscillator j. Taking inspiration from this finding, for a
phase shift is approximately 180°. We are also able to color
general
network
the network clearly
in two unbalanced
colors (color 1: vertex
0 and color with N oscillators, the compensating capacitance Ccomp,i for
2: vertices 1 and 2). In general, the compensation capacitor
oscillator
{0, 1,of. N. . oscillators
, N − 1is} is computed via
i
Ccomp
for the
oscillator i ini a∈network
given by:
i
Ccomp
= (nmax − ni ) · (CC ||C)
nmax = max nj
0≤j<N
(4a)
Ccomp,i
= (nmax − ni ) · (CC k C ),
(13)
(4b)
the number
oficouplings
for
with ni as where
the numberniofiscouplings
of oscillator
and
nmax as the maximum number of couplings, which at least
one of the N oscillators, the network is comprised of, has.
nmax
Hereafter, the balanced associated network will be used for
the implementation of graphs.
oscillator i, while
=
max {ni },
0≤ i < N
(14)
C. Impact ofisthe
device-to-device
variability
of theofNbO
the
maximum
number
couplings
x
memristor
per oscillator in the original network. While reducing
the
coupling
capacitance
may
allow
to accelerate the phase dynamics in the memristive
A major issue is the device-to-device variability of the NbOx
memristor and
its impact on the
behavior of
the network. of factors influence its selection. In this regard, to name but
computing
engine,
a number
In Fig. 6, the behavior of the graph in Fig. 1(a) and its
a couple
key
first
associated balanced
networkof(Fig.
5) isaspects,
shown taking
the CC should not be too small, otherwise the capacitive path
device variability
into account.
The dashed
shows, that theto be paired so as to reproduce the links, joining the vertices
between
any
two line
oscillators,
phase shifts of oscillator 1 and 2 with respect to oscillator 0 do
the
associated
graph,
across
the proposed cellular medium, would effectively act as
not converge.of
Thus,
the network
cannot be used
for the task
of
an open circuit. Concurrently, CC may not be so large as to violate the validity of the
approximation C >> CC , used to derive the compensating capacitance Ccomp,i for each
oscillator i ∈ {0, 1, . . . , N − 1} (refer to Equation (13)), which enables to counteract the
non-uniformity in the load capacitance per oscillator across the cellular medium, allowing
to keep the natural frequencies11 of the oscillators close together, which facilitates the
convergence of the memristive computing engine to some steady state.
In the remainder of this paper, any unbalanced network will first be compensated, and
then simulated for the solution of a given graph coloring problem.
3.3. Compensation for the Memristor Device-to-Device Variability
Simulating the network of Figure 5(b) for the case, where the device-to-device variability is taken into account, the memristor currents in the two oscillatory circuits may
be unable to settle on steady-state oscillatory waveforms. Figure 7(a) shows the currentvoltage loci obtained by simulating the memristor model Equations (1), (2), and (3) under a quasi-DC voltage stimulus for each value of the variability parameter α in the set
11
The natural frequency of an oscillator is the inverse of the period of the oscillations developing across
its circuitry.
J. Low Power Electron. Appl. 2022, 12, 22
11 of 30
{0, 0.2, 0.4, 0.6, 0.8, 1.0}. It is worth to pinpoint that the distribution of quasi-static characteristics, visualised in this figure, matches the variability observed in analogous loci measured
from 196 device samples. Assuming that the oscillator 0 (1) in the two-cell network hosts
a memristor, featuring a quasi-DC im –vm locus lying in the center (on the right end) of
the distribution of Figure 7(a), the network is found to fail to converge to steady-state
oscillatory dynamics. This issue is essentially due to the significant mismatch between
the DC operating points of the resistive nano-switches in the two oscillators. It may be
addressed by reprogramming the DC operating point of the memristor in cell 1. This may
be achieved [28] by adjusting either the resistance RS of the series resistor, as done in this
research study, or the voltage VS of the DC source within the circuit of oscillator 1 until an
anti-phase synchronisation pattern is found to emerge in the network.
As demonstrated in the phase diagram visualised in plot (b) of Figure 7, stepping the
increment ∆RS,1 in the series resistance RS of oscillator 1 from 0 Ω up to 151 Ω, the network
keeps featuring non-convergent phase dynamics for a while (see the orange trace relative
to the case ∆RS,1 = 50 Ω). Then, from some point onward, the memristor currents in the
two oscillators settle on steady-state oscillatory waveforms, sharing the same frequency,
but differing in phase by an offset, which progressively approaches the expected 180◦ level
as the series resistance RS of oscillator 1 gets larger (compare the green and purple traces,
obtained for the first and second ∆RS,1 -value in the set {100, 125} Ω, respectively). Finally,
the two capacitively-coupled cells attain anti-phase synchronisation (refer to the red trace
corresponding to the scenario ∆RS,1 = 151 Ω).
As another example, Figure 7(c), where the orange and green dashed traces illustrate
the time evolution of the phase of oscillators 1 and 2 relative to oscillator 0, respectively,
demonstrates that the balanced network of Figure 6(b) fails to converge to a steady-state
oscillatory solution when the first, second, and third value in the set {0.5, 0, 1} is assigned
in turn to the variability parameter α in the model of the memristor in oscillator 0, 1, and
2. A numerical procedure, tuning separately12 , one at a time, the series resistances in
(s)
oscillators 1 and 2 with the intention to maximise the steady-state phase shifts ϕ1 and
(s)
ϕ2 , determines that decrementing (incrementing) the series resistance of oscillator 1 (2) by
∆RS,1 = −134 Ω (∆RS,2 = +151 Ω), the network exhibits a steady-state oscillatory pattern
characterised by the anti-phase synchronisation between oscillators 1 and 2, on one side,
and oscillator 0, on the other side.
In the remainder of this section, in order to compensate for the negative effects that
the memristor device-to-device variability has on the performance of a balanced network,
the following approach shall be adopted. First, a reference cell, hosting the memristor, to
which the random number generator assigns a variability parameter value closest to 0.5
among all NbOx devices in the array, should be selected. In a hardware implementation of
the memristive network, the memristor of the reference oscillator would approximately
display the average dynamical behaviour among all the resistive nano-switches employed
in the array13 . This would minimise the subsequent adjustment to be carried out on the
bias circuit of each oscillator j ∈ {1, . . . , N − 1} to reprogram the DC operating point of the
respective memristor14 . Then, for each j-value in the set {1, . . . , N − 1}, the oscillator j is
capacitively coupled only to the reference oscillator 0, and the series resistance RS in its
12
13
14
In order to reprogram appropriately the operating point of the memristor in the cell j ∈ {1, 2} of the 3-oscillator
network under focus, the cell j itself is capacitively coupled only to the reference cell 0, and, as described earlier,
∆RS,j is tuned until anti-phase synchronisation emerges in the resulting two-cell network. This procedure is
carried out separately for oscillators 1 and 2.
It is important to pinpoint that, while the choice of a reference cell for the preliminary compensation of the
memristor device-to-device variability should fall for a specific oscillator, as specified here, no rule dictates the
selection of a reference cell for the later computation of the relative phase pattern of the array, as discussed in
Section 4.
It is important to observe that, while taking the proposed device-to-device compensation measure, care need
to be taken so as to keep the natural oscillation frequency of each oscillator within a close range. In fact, a wide
spread in this parameter, inevitably differing across the cellular medium, due to the RS tuning procedure,
would jeopardize the convergence of the bio-inspired computing engine to some steady state.
J. Low Power Electron. Appl. 2022, 12, 22
12 of 30
DC bias circuit is adjusted through a numerical procedure till the point when its increment
or decrement by ∆Rs,j induces a 180◦ phase shift between the memristor currents of the
two cells.
It is important to observe that, in hardware, such RS tuning procedure needs to
be carried out only once, during the computing machine testing phase, directly after
its fabrication. In order to simplify the programmability of the series resistor, it could
be implemented through a voltage-controlled CMOS transistor forced to operate in the
linear region.
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS
3
IEEE TRANSACTIONS
4 ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS
(a)
im /mA
3.0 2 90°
iM /mA
2.0
1.5
(c)
90°
90°
α = 0.0
α = 0.2
45°
α = 0.4
α = 0.6
3ms
α = 0.8 2ms
α = 1.0
1ms
135°2.5
1 2 180°
(b)
0ms
135°
∆Rs,1
∆Rs,1
∆Rs,1
∆Rs,1
135°
45°
3ms
0ms
180°
0° 0
12ms
180°
1ms2
=
=
=
=
2 90°
12
151Ω
45°
125Ω
100Ω
3ms
50Ω
135°
45°
2ms
3ms
1ms
2ms
0ms
0° 0
1ms
0ms
1 2 180°
0°
0° 0
1.0
225°
0.5
225°0.0
225°
315°
0.0
0.2
1 0.4
0.6
vm/mV
0.8
315°
315°
270°
225°
1.0
315°
1
270°
Fig. 4: Evolution
270° of the phase shift (referenced to oscillator 0) of the three
oscillators for the graph of Fig. 1 for the unbalanced (dashed) and balanced
Fig. 3: Quasi-static current-voltage characteristic of distinct realizations of
Fig.
6: the
Evolution
of the
(solid)
case without
variability
= 0) between
memristors.
The phase
bold shift (referred to oscillator 0) of the three
Fig.accurately
7: Phase shift
behavior
for two
coupled(αoscillators
with
variability
for
Fig. 6: Evolution of the
thememristor
phase shift
(referred on
to the
oscillator
0) parameter
of the three
M depending
variability
α, which
numbers
the numbering
the
respective
vertices.
oscillators
for
the graph of Fig. 1(a) with variability (oscillator 0: α = 0.5,
distinct
adjustments
∆Rrepresent
of the series
resistor of
RS
. The
reference
oscillator
oscillators for the graph
of Fig.
with variability
(oscillator
0: 196
α =NbO
0.5,x-based
reflects
the 1(a)
distribution
obtained by
measuring
threshold
1: α = has
0.0 and oscillator 2: α = 1.0) between the memristors.
(blue line) has the variability parameter α = 0.5, theoscillator
other oscillator
oscillator 1: α = 0.0switching
and oscillator
devices.2: α = 1.0) between the memristors.
270°
Figure 7. (a) Spread in the distribution of quasi-DC memristor current-voltage loci, as emerging
Depicted
is the all
behavior
of eachof
oscillator
the uncompensated (dashed
α = model
1.0.
Depicted is the behavior
of each
oscillator for thesimulations
uncompensated (dashed
from
numerical
of the
Equations (1), (2)C,Cand
(3) for
values
the for
variability
C
C and compensated (solid line) cases. In the compensated case, oscillator
line)
line) and compensated (solid line) cases. In the compensated case, oscillator
1
was
adapted
by
∆R
=
−134Ω
and
oscillator
2
by ∆R = 151Ω. The bold
1 was adapted by ∆R parameter
= −134Ω and oscillator
by ∆Rset
= 151Ω.
bold
α in2 the
{0,The
0.2,
0.4, 0.6, 0.8, 1.0}. (b) Phase diagramnumbers
showing
the time evolution
of the
are the labels identifying the vertices of the graph under study.
numbers are the labels identifying the vertices of the graph under study.
CC ||C
CC ||C
B. Unbalanced number of connections
◦
11 the0 reference
phase shift of oscillator 1 relative to the 0 -valued2phase of
oscillator 0 for each of the
2
If the graph of Fig. 1(a) will be implemented with its associvalues of
the
seriescanresistance
increment ∆R
in the set {50, 100,
125,
151}Fig.Ωby
(refer in turn to the
Fig.S,1
5: Balanced associated network of vertex
the graphcoloring.
depicted inBut
1. a small adjustment of the DC operating
atedbynetwork
Fig. 1(b), we
determine
the evolution of
vertex coloring. But
a smallinadjustment
of the DC
operating
the
phase
shift
between
the
oscillators,
as
depicted
in
Fig.
4
point
of
each
oscillator,
the characteristics of the
3
0
orange,
green,
andofred
achieveonanti-phase
point of each oscillator,
depending
on thepurple,
characteristics
the traces). The two capacitively-coupled oscillators depending
by dashedthis
lines.
For can
this be
phase
shift In
oscillator
corresponding memristor, this issue can be solved. In Fig. 6
corresponding memristor,
issue
solved.
Fig. 6 0 will serve
forshift
therelative
largest
this set. (c) Phase
diagram
illustrating
the
assynchronisation
reference and a large phase
to this∆R
reference
S,1 -value
the solid
lines show
the behavior of
thephase
network with adsince Cin
C C, with:
the solid lines show
the behavior of the network
with adwill be interpreted as a different color. On the radial axis, the
in the resistance
of the
series
resistor
5 balanced
4
CChoc
C changes
hoc changes in the
resistance ofoftheoscillators
series resistor1 R
dynamics
(in
orange)
and
2
(in
green)
for
the
network
of
Figure
6(b)
for
the RS . As
S . As
time is shown and the angle in this plot illustrates the phase
. oscillators 1 and 2 are found to feature a phase shift of
CC ||C =
desired,
desired, oscillators
1
and
2
are
found
to
feature
a
phase
shift
of
C
+
C
C
shift.
The bold numbers
represent
the number of the
respective
case
specific
parameters
the modelFig.of8: the
inapproximately
the cells 0,180°
1, and
2 aretorespectively
Graph memristor
of a ring of six vertices.
with respect
oscillator 0. With regard to
approximately 180°
withIfwhere
respect
oscillator
With
regard toby in
vertex.
we hadtocoupled
only0.two
oscillators
a capacitor, The difference in this effective capacitance is approximately
strategy
to
find
the
right adjustment of the operating point
our strategy to find
thesystem
right adjustment
ofTable
the
operating
point
difference inwithin
the number
of connections
times
CC
||C.
If we
with
parameters
of
I, wesecond,
would
have and
recorded
a theα-value
controlled
by the
first,
third
the
setour
{
0.5,
0,
1
}
(see
the
dashed
traces),
of an oscillator, consider the simple case where two oscillators
of an oscillator, consider
the of
simple
case where
twobetween
oscillators
phase shift
approximately
180°
them, as known add this difference to the oscillator of the associated network
are
coupled.
The
first
oscillator
serves
as
a
reference and is not
and
the[15]–[18].
negative
the be
network
performance
to the
memristor device-to(as illustrated
Fig. 5),
theassociated
becomes
balanced.
on the
finalinphase
shifts
thenetwork
oscillators
feature
are coupled. The from
first
oscillator
as
a reference
and
is noton
[1],after
[8], serves
[9],
Thus,effects
similarly,
it should
would
be based
adjusted.
This reference
oscillator should feature a quasi-static
The evolution
of the after
phase the
shiftnetwork
for
the balanced
associated
within
the
associated
network
reaches
a
adjusted. This reference
oscillator
should
feature
a
quasi-static
desirable
to
achieve
a
phase
shift
of
180°
between
oscillators
device variability
have
been0 compensated
by incrementing
(decrementing)
the
series
resistance
S by
current-voltage
which lies inRthe
center of the
is depicted
in Fig.are4 (solid
lines).
As expected,
the
0 and 1 as well
as between
oscillators
and
to obtain
state. network
Thereby,
the vertices
sorted
according
tocharacteristic
current-voltage characteristic
which
lies in the
center of
the 2, stable
in Fig. 3. In our case we select a device
shiftsolid
is approximately
180°.
We
areassociated
also depicted
able to color
a ∆R
reproducible
maximum
separation
134
Ωphase
(∆R
+between
151 Ω)
(referofphase
to
lines).
thedifferent
values
thethe
steady-state
phase shifts
ofdistribution
their
distribution depicted
inS,1
Fig.=
3. −
In
our case
we select
a device
S,2 =
the network
clearly
in two colors
(color
vertex
within1:
α ascending
= 0.5.0 and
The color
other oscillator is adjusted by adapting
colors
independently
of adjusted
the chosen
But, as
seen in relative
oscillators
to the
reference
oscillator
with α = 0.5. The
other
oscillator is
by graph.
adapting
2: vertices
1 and 2).itInis general,
the
compensation
capacitor
Fig.by4 ∆V
(dashed
lines)the
we series
derive resistor
approximately
50° phase
shift. order.
theto
voltage
source
by ∆V or/and the series resistor by ∆R
numerical
Additionally,
necessary
provide
the
the voltage source
or/and
by ∆R
i
C
for
the
oscillator
i
in
a
network
of
N
oscillators
is of the
This
is
the
case
because
oscillator
0
has
two
connections
and
Remark
In view of
a future
implementation,
parameter
memristive
comp
until the phase
shift
to approximately
180°, as seen
information regarding
which vertices the
are connected.
This setting
is converges
until the phase shift
converges to2.
approximately
180°,
as seen hardware
oscillators 1 and 2 only have one. Because of this, there is given by:
inA.Fig.
7N
for×an
oscillator with α = 1.0 and the reference
defined
by
the
symmetrical
adjacency
matrix
This
N
in Fig. 7 for an oscillator
with
α
=
1.0
and
the
reference
i
could
becapacitance
optimized
to
increase
the
data
processing
speed
further.
However,
a
ancomputing
imbalance in theengine
network. The
effective
Cef
i
f
(blue line) (4a)
with α = 0.5. It can be determined,
Ccomp = (ngraph
noscillator
· (C
the whole undirected
vertices.
oscillator (blue line) with α = 0.5. It can be determined, matrix defines
max −of
i )N
C ||C) The
loading
the oscillator i ∈investigation,
{0, 1, 2}, consists of aimed
the internal
that
formatrix:
a too
small
adjustment
(∆R
=
50Ω) there is no
comprehensive
to
ensure
this
would
not
impair
the
accuracy
of
the
engine
following
expression
describes
the
elements
of
the
that for a too small
adjustment
(∆R
=
50Ω)
there
is
no
n
=
max
n
(4b)
max
j
capacitance C of the oscillator and in parallel to this, for each
0≤j<N
convergence. By implementing a larger change (∆R = 100Ω,
(
convergence. By of
implementing
a larger
change
(∆R =oscillators,
100Ω,
the physically-connected
neighboring
the
calculations,
should
concurrently
beseries
carried
out. On
onei ishand,
the
of the computing
coupled
to joperating
∆R
=of125Ω),
the speed
system
with 1ni ifasoscillator
the number
of couplings
oscillator
i and converges but not to a 180° phase
∆R = 125Ω), theconnection
system converges
but
not
to
a
180°
phase
between the respective coupling capacitorACc
and
Aji =
(5)
ij =
nmaxmemristor
as iftheoscillator
maximum
number
ofshift.
couplings,
which
at leastit
By
implementing
∆R
= 151Ω the system
engine
should
not
be
so
large
to
prevent
the
to
respond
to
the
stimuli,
experiences
over works as
0
i
and
j
are
not
coupled,
shift. By implementing
∆R
=
151Ω
the
system
works
as
the total capacitance loading the respective coupling capacitor
one of the N oscillators, the network
is comprised
of,athas.
intended
and stays
180° phase shift. However, there is a
1 phase shift. However, there is a
intended and stays
at
180°
Cc
itself.
Thus, we would
derive thenot
following
equations
for the thoroughly its rich dynamics
time,
which
allow
to
exploit
for
solving
the
challenging
whereby
i and
j
denote
the
column
and
the
row
of
the
Hereafter, the balanced associated variability
network will
be usedthe
foradapted and the reference oscillator.
between
three
thisthe
network:
variability between
theoscillators
adapted of
and
reference oscillator.
adjacency matrix.
As an example,
for the graph
a ring of sixhas been implemented for all oscillators in the
the implementation
of graphs.
This
adaptation
coloring
underinfocus.
On asthe
other
hand,
the
array
ofof capacitively-coupled
memristor
This adaptation hasvertex
been implemented
fortask
all oscillators
the
vertices
shown
in
Fig.
8,
we
derive
the
following
adjacency
0
network individually utilizing the same reference oscillator. If
network individuallyCutilizing
the2 ·same
reference oscillator. If
(CC ||C)
(3a)
ef f = C +
matrix
A:
oscillators
is expected to attain
a steady state within a sufficiently-short
time
all the
the
NbOxframe.
thevariability
system isof oscillating
faster, dueFor
to smaller
capacitors C
1
the system is oscillating
faster, due to smaller capacitors
CCC ||C (3b) C. Impact of the device-to-device

Cef
≈ C+
f = C + CC || (C + CC ||C)
memristor 0 1 0 0 0 1 and CC , it becomes difficult or even impossible to determine
and CC , it becomes
difficult
or
even
impossible
to determine
case
studies,
investigated
in
this
research
work,
except
for
those
simulations,
in
which
the
proposed
2
Cef
=
C
+
C
||
(C
+
C
||C)
≈
C
+
C
||C
(3c)
C
C
C
f
A major 
issue
the NbO
operatingofpoint
at which
the system works. In this case
1 is0 the1 device-to-device
0 0 0
x
the operating point at which
the system works. In this case

the variability
computing
engine
waswhich
unable
transient
phase,
the
value
assigned
to has
theto be
coupling
and

memristor
its impact
on the
behavior
of the
network.
device-to-device
variability
decreased, which is a
the device-to-device
variability has to
be decreased,
is a to exit the
0 1 0 1 0 0the
A
=
.
(6)
In Fig. 6,0the 0behavior
ingoal
Fig.in1(a)
andfabrication.
its
1 For example, for oscillator 1, which has only one physically-connected
graph
major
device
Hereafter,
all memristors
will
1 0 of1 the0to
major goal in device
fabrication. Hereafter,
all memristors
will


capacitance
C
enabled
the
oscillators’
phases
to
converge
steady-state
values
within
tens
of
C
neighboring oscillator, i.e. the reference oscillator 0, the total capacitance
associated0balanced
is shown taking
with athe
device-to-device variability by distinct
0 0 network
1 0 (Fig.
1be5)implemented
be implemented with
device-to-device
byparallel
distinct
loadinga the
respective coupling variability
capacitor is the
between the internal
dashed
line shows,
that
milliseconds.
Each0 design
maydevice
in variability
fact1 affect
some
figure
of the
thecorresponding
bio-inspired
parameters
αof∈merit
[0,
1]the
and
oscillators will
0 into
0 account.
0 1 The
0key
the reference
oscillator
and
the series parameter
connection
between the
parameters α ∈ capacitance
[0,
1] andof the
corresponding
oscillators
will
phase shifts of oscillator 1 and 2 with
to as
oscillator
0 doin this section.
capacitance, which couples oscillators 0 and 2, and the internal capacitance
be respect
adapted
described
be adapted as described
in2.this section.
For example, reducingAccording
the capacitance
of each
oscillator
may
converge.
the coloring,
network
cannot
used
for induce
the task of an acceleration in
ofnetwork.
oscillator
tonotthe
rules C
ofThus,
vertex
only be
unconnected
vertices are allowed to have the same color. This means that
III. G ROUPING
VERTICESNDR
INTO COLORS
the phase
dynamics.
However, it
could also reduce the range of admissible
memristor
bias
III. G ROUPING
VERTICES
INTO COLORS
for all pairs of vertices m, n in a color Ck , the corresponding
One
of
the
most
significant
parts
of
the
vertex
One of the most significant
of thewhich
vertex coloring
prob- element
points,parts
about
the processing
unit
would
undergo
limit-cycle
oscillations
[44],
which,
ascoloring
a probAmn of the adjacency matrix has to be zero:
lem is the decision regarding which vertices belong to the same
lem is the decision regarding which vertices belong to the same
consequence,
would
shrink
the tuneable range
series
R(7)S be
inassigned
the device-to-device
∀m,for
n ∈ Cthe
. group and should
to the same color. This decision
k : A
mn = 0resistance
group and should be
assigned to the same
color. This
decision
variability compensation procedure.
4. A Rigorous Strategy for Coloring a Graph via the Network Phase Dynamics
As explained earlier, the assignment of colors to the nodes of a graph is based upon the
relative phases among the oscillators of the associated network at steady state. However,
a rigorous strategy, as proposed below, needs to be applied at the end of the network
J. Low Power Electron. Appl. 2022, 12, 22
13 of 30
simulation to classify the vertices of the associated graph into color groups, with the
intention to use the minimum possible number of colors15 .
As described earlier, the relative phases among the oscillators of a given network are
computed at the end of a simulation, on the basis of the time instants at which, within a
common steady-state period, the time waveforms of the currents through the memristors
cross a threshold value Ith , here set to 0.5 mA, during the ascending phase. This calculation
allows to order the relative phases in increasing order, with the 0◦ reference level sitting
on the first position of the arrangement, which we refer to as phase shift ordering in the
remainder of the paper. The phase shift ordering directly translates into a corresponding
ranking among the oscillators, with those, which feature a lower steady-state phase relative
to the reference cell, sitting higher in the table. Equivalently, the ranking among the
oscillators may be interpreted as a ranking among the associated vertices.
Our rigorous strategy to assign colors to the vertices of the associated graph on the
basis of the network phase dynamics is based upon the analysis of the oscillator/vertex
ranking through an iterative procedure composed of N iterations. The first iteration
may be summarised as follows. Initially the first vertex in the table, i.e., reference vertex
0, is inserted in the first color group. Proceeding toward the bottom of the table, for
j ∈ {2, . . . , N }, vertex at position j in the table is assigned the same color as ( j − 1)th -placed
vertex if the graph features no edge between these two vertices, otherwise the jth -ranked
vertex is defined as the first element of a new color group. After coloring vertex at row N in
the table, a final check needs to be carried out to verify if the vertices in the last color group
may be merged with those in the first color group. This may be done if and only if no pair
of vertices in these two groups is connected by means of an edge in the undirected graph.
In cycle i ∈ {2, . . . , N } of the iterative procedure, the color assignment step is repeated in a
similar fashion, analysing progressively the vertices at positions i, i + 1, . . ., N, 1, 2, . . ., i − 1
in the table. With such iterative procedure, at least one of the cycles will allow to determine
the minimum number of color groups, identifiable by the network, given the phase shift
ordering it outputs at the end of a certain simulation. In other words, indicating the kth
color group, which, on the basis of the prediction of the ith cycle of the iterative procedure,
(i )
is identifiable by the network, as Ck (k ∈ {1, . . . , mi }, where mi ∈ [n, N ] denotes the total
number of colors assigned to the N nodes of the associated graph in the ith iteration, and n
represents the chromatic number of the graph itself, the proposed strategy will output the
N
particular group classification obtained from the qth iteration, whereby mq = minii=
=1 { m i } .
Remark 3. Remarkably, the initialisation of the oscillatory network, and, particularly, the temporal
order of activation of the DC voltage sources in its oscillators, plays a crucial role on the steady-state
phase arrangement, and, as a result, on the outcome of the proposed strategy. Consequently, it
is possible that the classification of the vertices of a graph into color groups, as estimated via our
iterative procedure, is not optimal. Under these circumstances, the network is unable to identify the
chromatic number of the associated graph, since it converges to an oscillatory solution corresponding
to a local minimum for some optimisation goal associated to the vertex coloring problem.
Let us denote the phase shift vector as ϕ , [ ϕ0 = 0◦ , . . . , ϕ N −1 ], where ϕi stands for
the phase shift between oscillator i and reference oscillator 0 over the course of a certain
simulation of the network (i ∈ {0, . . . , N − 1}). Given that the network of capacitivelycoupled oscillators tends to pull the phases of physically-connected cells far apart one
15
The proposed strategy will determine the minimum possible number of color groups, which, under a given
initialisation setting, the network is able to identify as it classifies the nodes of the associated graph. Importantly,
as will be clarified later, this minimum number does not necessarily coincide with the chromatic number of
graph, since the network may converge to a correct but suboptimal solution. Methods allowing the memristive
array to overcome a suboptimal solution so as to approach the optimal one will be presented shortly.
J. Low Power Electron. Appl. 2022, 12, 22
14 of 30
from the other, it is expected that the phase dynamics unfold toward a steady-state pattern
maximising a function F (ϕ) of the form
F (ϕ) ,
1 N −1 N −1
·
∑ ai,j · | ϕi − ϕ j |,
2 i∑
=0 j =0
(15)
where ai,j is the element at row i and column j of the so-called adjacency matrix A, which
encodes the coupling arrangement within a N-node graph16 . Transforming the expression
for F (ϕ) in Equation (15), another function of the phase shift vector, defined as
G (ϕ) ,
1 N −1 N −1
·
∑ ai,j · cos( ϕi − ϕ j ),
2 i∑
=0 j =0
(16)
is chosen to describe the optimisation goal of the network as it evolves toward a stable
(s)
(s)
solution. It is worth pinpointing that the phase shift vector ϕ(s) = [ ϕ0 = 0◦ , . . . , ϕ N −1 ],
emerging in a well-behaved network at steady state, minimises the function G (ϕ). For a
general network, however, the optimization goal function might feature a number of local
minima besides the global minimum. If the oscillatory solution, the network converges to,
at the end of a certain simulation, corresponds to a local (the global) minimum of G (ϕ),
the application of the vertex coloring strategy, described earlier, results in a group number
higher than (equal to) the chromatic number of the associated graph.
The analysis of an exemplary network, specifically the one shown in Figure 3(b),
allows to gain a deeper insight into the phase dynamics of a memristive array in a pair of
simulation scenarios, associated to a suboptimal and to the optimal initialisation scenario,
respectively. The network under focus is already balanced, since each of its oscillators is
coupled to the 2 adjacent cells, thus only the memristor device-to-device variability needs
to be neutralised. Figure 8(a) ((c)) visualises the time evolution of the phase dynamics
emerging in the network under a suboptimal (the optimal) initialisation setting. In both
scenarios the optimisation goal function decreases over time, but, in the first (latter) case,
G (ϕ) converges asymptotically toward a local (the global) minimum, as depicted in plot (b)
((d)) of the same figure.
16
If an (no) edge connects vertices i and j, then ai,j = a j,i = 1(0). Note that A = AT ∈ R N × N .
J. Low Power Electron. Appl. 2022, 12, 22
15 of 30
SYSTEMS—I: REGULAR PAPERS
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS
5
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS
(a)
6
(b)
G(ϕ)
G(ϕ)
G(ϕ)
not several vertices can be
90°
41 Starting the routine
an be performed
utilizing
90°
onebyvertex
after the other as starting point.
2
1
−1.6
cy matrix A, which
includes
from vertex
1, the first test would 135°
check whether or not 45°
vertex
−1.8
135°
45°
10ms
8ms
onging to the tested
4 mayvertices.
be included in the same color group C1 as 6ms
vertex
1.
3
0
−2.0
10ms
4ms
8ms
ted in the sameWith
colorone
Ck ,ofthethe phase shift vector elements as starting
−2.2
point,
2ms
6ms
0ms
4ms
180°number of colors,
0°
o matrix.
2ms 4
54
−2.4
the identification of the minimum
which 03the
0ms
1 180°
0° 2
0
−2.65
network
is able to identify with such a phase shift ordering, is
scillators of the
associated
3
−2.8
Note
of the verticesensured.
according
to that this method, referred to as standard in the
−3.0
remainder
of the paper, does not 225°
guarantee the determination
Physically-coupled
oscillators
315°
25 implemented by the
4
6 315°
0 225°
2
8
10
of oscillations.
the chromatic number of the graph
ft between their
270°
SYSTEMS—I: REGULAR PAPERS
6
t/ms
However, the number of sub-matrices which need
270°
ller phase shiftnetwork.
are probably
9: Exemplary
trajectories
for the(c)phase
of the associated
(d) G(ϕ) for the phase evolution of
Fig. 11: network
Evolution of the minimization goal
o belong to the
group isFig.still
to same
be analyzed
much lower
compared
to theshifts
number
for the graph of a ring of six vertices shown in Fig. 8. All capacitor
voltages
Fig.
9,
which
trends
towards
a
local
minimum
of −3 (compared to the global
phase
wethe
derive
the
of
possible
orderings
the vertices,
i.e. in(Nthe
−network
1)!, obtained
and allof
memristor
temperatures
are initialized
respectively
to obtained in Fig. 10(b)). The assignment of vertices to color
90°
ng
point.shifts,
Starting
routine
minimum
of −6
−2
2
1
0V
and
to
T
=
293K.
The
voltage
of
the
DC
source
in
each
cell
is
systematically
be
tested
for
any presorting according
to the phase shift. If the groups, on the basis of the final phase shifts of the
amb
respective oscillators, is
check whetherwithout
or not vertex
135°
45°
ramped
from
0V
to
V
over
a
time
of
1µs.
A
random
sequence
is
generated
S
cientlygroup
decreases
number
visualized in Fig.−3
9 by highlighting the perimeter of the portion of the circle,
network
works well, the
sorting will lead to theat which
color
C1associated
asthe
vertex
1.
to dedicate a particular ordering in the time instants
the N DC
3 trends of the0 phases of the
10ms
where the 2 trajectories, indicating the temporal
reasonable
calculation
time.
8ms
minimal
number
of
colors
for
this
graph.
voltages begin to rise.
r elements as starting point,
6ms
oscillators corresponding to the vertices 0 and 3, 1 and 4, 2 and 5, respectively
4ms
to
check of
allcolors,
possible
sortings
−4state, with the same colors used in this plot to fill the two
4 belong to at steady
2ms
number
which
the
4
5
0ms
1 180°
0° 2 circles symbolizing the corresponding vertices of the
N
vertices,
this
would
result
graph. This is a correct
5
IV.
T
HE
NETWORK
AS
OPTIMIZATION
ALGORITHM
0
uch a phase shift ordering, is
3 the corresponding sub-matrix Asolution
−5 coloring problem, but not the optimal one, since it does
of the vertex
mple
a network
with Nin=the
25 upon analyzing
1:
eferred
to as standard
In this section
we will gain a deeper insight into the operation
not provide the minimal number of colors, i.e. the chromatic number of the
23
10
possible
combinations.
guarantee the determination
of the whole network. Here we interpret
the
A
A01system 0as 1an graph under consideration.
−6
ding
to implemented
the phase shifts,
we
A1 =225° 00
=
.
(9)
graph
4
6
0
2
8
10
engine by
for the
optimization algorithm.
We A
will
see
that, due315°
to 0an
A
1
10
11
forsub-matrices
the optimal coloring.
For
of
which
need
70°
2
t/ms
improper
choice of the initial conditions, the system may be
thecompared
ordering to
of
the number
vertices
Because the sub-matrix A1 is not equal to the zero matrix,
er
the
unable
to
identify
the minimal number of colors. This can be properties, for instance the device-to-device variability, which
ssociated
oscillators,
we
first
vertex
0 and 1 cannot be grouped in the same color.
ces, i.e. (N −interpreted
1)!, obtained
Fig. Thus,
10: (a)the
Evolution of the phase shift of the associated network for the
as
local
solution of
an optimization
problem,
8. (a)
Phase
dynamics
the{0}
balanced
ofassignment
Figureof3(b)
with
compensation
for
the
−2assigned
2 where
differently
operation
each
oscillators,
the turn
i.e. Ifto the
the first
color isFigure
of ((c))
vertex
0 only,
i.e. C1 1ofaffects
=
while innetwork
graph
depicted
Figthe
8. The
of vertices
to color groups,
onon
the
g vertex/oscillator,
to the phasethe
shift.
solving
algorithm
reaches
its
limit.
In
the
following,
two
basis
of
the
final
phase
shifts
of
the
respective
oscillators,
is
visualized
in
sequence
of
the
voltage
sources
of
the
oscillators.
Because
the
memristor
device-to-device
variability
and
under
a
suboptimal
(the
optimal)
initialisation
setting.
(b)
shift,
in
the
ordered
list.
We
we have to create a second color, which is associated with
the sorting willpossible
lead tosolutions
the
−3 local solution and 3reach the 0plot (a) by highlighting the perimeter of the left (right) half circle, which the
to avoid ((d))
the
optimization
goal
G(ϕ)
in
equation
(12)
is
a
non-convex
funccond
vertex/oscillator
of
this
Timestep,
evolution
of to
thetest
optimisation
goalalso
function
a local
(theofglobal)
minimum
in the
vertex 1. In the next
we have
if vertex
4 can
s graph.
3 trajectories,
indicatingtoward
the temporal
trends
the phases
of the oscillators
global
be explained
and validated.
color to it, unless
thesolution
vertex will
may
colored
with the second
color. Because vertex 1 tion,
and 4itare
notfeature several local minima. The one with largest
corresponding
the first
vertices
1, 3, and
(0, 2,application
and 4), belong
at vertex
steady
simulation
scenario illustrated in plot
(a) ((c)). Intothe
(latter)
case5 the
of to
the
−4
4
5
state, with the
colors used
plot (b) to
fill the
three circles
symbolizing
pled
to the reference oscillator connected i.e.
modulus
is same
the
global
The
others
only
local
the sub-matrix
Athe
2 : respective vertex
MIZATION ALGORITHM
coloring
strategy to
ranking
divides
theoptimum.
6 in
nodes
ofEvolution
the
graph
ofare
Figure
3(a)
into
the associated
verticesoptimization
of the graph. (b)
of
the gradient
minimization
goal
goal
of the
solutions.
−5 of theoptimization
zing the respective
sub-matrix representation
A. Mathematical
Simple
algorithms
like
based
3
(2)
colors.
In
the
first
(latter)
case
the
composition
of
each
of
the
3
(2)
color
groups
is
made
clear
by
G(ϕ)
for
the
vector
phase
evolution
shown
in
(a),
which
trends
towards
the
per
insightIninto
the
operation
A11 A14
0 0
le later).
thenetwork
case
there is
methods
and also
theThe
proposed
implementation
of the
vertex
A2 =
=
(10)
global
minimum
of
−6.
initial conditions
of allthe
capacitor
voltages
and
interpret
the
system
as
an
A
A
0
0
the
colors
assigned
in
plot
(a)
((c))
to
the
arcs
of
the
circular
sectors,
which
host
final
destinations
−6
41
44
two vertices, the
one
memristor task,
temperatures
in the network
are set asdynamics
reported in of
the coupled
caption of
coloring
exploiting
the analogue
Thesecond
oscillatory
interpreted
as an
0 traces
2 associated
4 engine
6 to solving
8
10 clustering together, as well as by the colors assigned in the
.group.
We will
see the
that,
due
to an network can be
of
the
phase
shifts
Fig.
with
a different stop
random
to dedicateand
a particular
Then
third
vertex
memristor
oscillators,
at sequence
local solutions,
remainordering
there
is the zero
matrix,
they
cantime
both
be grouped
in
the9 but
second
/ ms
optimization
problems.
We
know
that,
under
the
given
condiinnodes
the time
at which
thethe
N DC
voltages begin
to rise.
nditions,
system
may
be
inset 4}).
of plot
(b)next
((d))vertex
to the for
respective
of instants
the
Since
chromatic
number
of the
graph
group
hasthe
been
created
so capacitively
far,
color (C
The
testingafterwards,
is vertex
2,
forgraph.
improperly
chosen
initial conditions.
There
areis
2 = {1;
(b)
tions,
two
coupled
oscillators
maximize
the
phase
mber
colors.
This
can beto
colorofonly
if it is
coupled
2,
the
relative
phases
among
the
oscillators
of
the
network
converge
to
a
suboptimal
(the
optimal)
which
is
not
connected
to
vertex
4
but
is
coupled
to
vertex
some algorithms
which are able to overcome local solutions,
Fig. 10:
Evolution
the are
phaseable
shift to
of generalize
the associated network
for the
shift between them
to (a)
180°.
Thus,ofwe
optimization
on the otherproblem,
hand, twowhere
color graph
depicted
in Fig2 8.cannot
Thethe
assignment
of vertices
to color
groups,
on
the
pattern
in
simulation
of
(a) ((c)).
1. Thus,
vertex
be
included
in plot
the
second
color
and
leading
to
the
global
solution. Inspired by distinct approaches,
this
for
a
network
of
coupled
oscillators.
For
a
network
of
limit.
In vertex
the following,
twoa basis of the final phase shifts of the respective oscillators, isFig.
visualized
in
8. toThanks
to the [27]
correct
initial conditions,
we are
the
third
is assigned
will
be
assigned
a
third
color.
Vertex
5
can
be
added
the
like
genetic
algorithms
or simulated
annealinghere
procedures
N oscillators
adjacency
matrix
A, we assume
plot (a)
by highlighting
the perimeter
of the left that
(right)the
half circle, which the
ocal
solution
reach
the with
coupled
to theand
second
vertex,
able
to determine
that
theproposed
oscillators
can be
divided
into
two
Let
us
gain
a
deeper
insight
into
the
outcome
of
the
graph
coloring
strategy
third
color
(C
=
{2;
5}).
Finally,
vertex
3
has
to
be
associated
[28],
two
possible
strategies
to
determine
the
global
solution
3
3
trajectories,
indicating
the
temporal
trends
of
the
phases
of
the
oscillators
system maximizes the sum of the phase differences between
andcolor
validated.
me
group of the second corresponding
groups,
associated
to
two
different
colors
respectively.
The
first
to
the
vertices
1, of
3, and
5is
(0,
2, and 4), belong
tosolving
at 2steady
with
a
fourth
color,
because
it
connected
to
vertex
and,
for
the
case
the
suboptimal
simulation,
which,
as
demonstrated
in
Figure
8(a),
outputs
for
the
vertex
coloring
problem
via
coupled
oscillators
coupled oscillators given
by:
the
same
colors
usedassigned
in plot (b)the
to fill
the three
circles
symbolizing
outine goes on in a similar state,
color
ispresented
assigned
to sections
all even IV-C
vertices
2, 4) and the second
as awith
result,
may
not be
third
color.
Since
the
first
will
in
and(0,
IV-D.
the
steady-state
shiftbe
vector:
X
Xfollowing
the
associated
vertices
of the graph.
(b) Evolution ofphase
the minimization
goal
1
fhetheremaining
optimization
goal of
vertices
in the
themax
color
to
all
odd
vertices
(1,
3,
5).
For
rings with an even
oscillator
and
the
oscillator
with
the
highest
phase
shift
could
G(ϕ) for the vector phase
in (a), which
Aijevolution
|ϕi − ϕshown
(11) trends towards the
j|
one. To illustrate these steps, ϕglobal
minimum
of
−6.
The
initial
conditions
of
all
capacitor
voltages
and
(
s
)
(
s
)
(
s
)
(
s
)
(
s
)
(
s
)
number
of
vertices,
the
minimal
number
of colors is always
be2 close
to
each
other
in
the
phase
plot,
a
final
test
will
be
(
s
)
T
◦
◦
◦
◦
0≤i<N 0≤j<Nϕ
=
[ ϕ}0are, setϕ1as ,reported
ϕ , ϕ3the, caption
ϕ4 , ϕof5 ] = [0 , 118 , 240 , 358 , 120◦ , 242◦ ]T . (17)
temperatures
the to
network
|
{z inout
ng of sixasvertices
depicted
C.
Obtaining
erpreted
an engine
solvingin memristor
two.
Thus,
theglobal
correctsolutions
solution by
cancrossover
be determined utilizing the
additionally
carried
verify
if the first2and inthe
last
vertex
Fig. 9 but with a different
=F (ϕ) random sequence to dedicate a particular ordering
hase under
shifts the
of the
associated
that,
given
condi- ininthethe
can
be grouped
the same
color
C1 . first
Because
properly
initialized snetwork.
10(b)
s evolution
timeorder
instants
at The
which
the N DCinvoltages
to rise.
The
possibility
of ourto
resulting
phasebegin
shift
ordering,
namely
ϕ0to−overcome
ϕ1s − Fig.
ϕ4s −the
ϕ2slocal
−shows
ϕ5ssolutions
− ϕthe
3 , allows
9. Thesemaximize
trajectories
in vertexthe
cillators
theresult
phase
3 isoptimization
not connected
to vertex
vertex
3 can
be
inserted
of
the
optimization
goal.
The
value
of
the
optimization
goal
where
we
highlighted
goal
F (ϕ). 0,
Here
ϕ is
17
optimization
goal G(ϕ) is inspired by genetic algorithms.
establish
the
following
vertex
ranking
:
s,shifts:
we are able the
to generalize
color Cof
procedure results
a grouping
the
vertices
G(ϕ)
first
decreases
exponentially
and
then
asymptotically
1 . This
vector of allinphases
the oscillators
and ϕin
the ith of
i is
Genetic algorithms simulate the evolution of a population,
scillators. For element
a network
into
three
C1 =
C2 = individuals
{1; 4}
converges
to approximately
−6,generation
which is the
minimum
of of
ϕ and
hence
thedifferent
phase
ofcolors,
the
ithnamely
oscillator.
Aij{0;is3},0where
for3.the next
are global
generated
from
Fig.
8.
Thanks
to
the
correct
initial
conditions,
here
− 1inwe
−Fig.
4are
−8.2 − 5 −
(18)
th
th
trix A, we assume
that
the
and
C
=
{2;
5},
for
the
example
of
the
graph
shown
3 row and j
of
the
optimization
problem
(12)
for
this
graph.
the
element
in
the
i
column
of
the
adjacency
>
the
old
population
by
mutation
and
crossover
and
only the
able to determine that the oscillators can be divided into two
ϕ3
ϕ4
ϕ5 )
e phase differences
between
In this case
the testing routine,
from
vertex
0, resulted
◦
matrix
the optimization
goal Fstarted
(ϕ),
will
fittest
individuals
of the old population and the new individuals
groups, associated
to two different
colorswe
respectively.
The
first2
58◦ 120◦ 242
)>A.
. To
(8) normalize
17 Since,
here,
oscillator
i
is
associated
to
vertex
i for each
in
the
indication
of
three
different
colors
for
this
graph
. i ∈ {0, . . . , N − 1}, the oscillator sequence, correspondtake the cosine function
of each phase
difference,
which
leads
will
be
selected
for
next generation. This means that the
color is assigned
to
all
even
vertices
(0,
2,
4)
and
the
second
Local and
globalthe
solutions
to the phase
maybe
beB.
indifferently
However,
for aing
different
graphshift
thisordering,
might
not
the
case andreferred to as oscillator ranking or vertex ranking. As will
to the change from
a maximization
to
a
minimization
problem
fittest
individuals
would
the of
ones
forming
population
color
to
all
odd
vertices
(1,
3,
5).
For
rings
with
an
even
be clarified later on, this is not alwaysDepending
the case, when
actions
are
on
the
network
Aij |ϕi − ϕj |
(11)
the initialbestate
theperformed
system, the
sometimes
theto
therefore
theoptimization
testing
routine
to be repeated by
selectingon perturbation
and
results
in
the
following
task:has number
enhance
its
performance.
number
of
vertices,
the
minimal
of
colors
is
always
with
the
best
score
in
some
health-monitoring
measure
(i.e.
the
he
vertices
is:
0;
1;
4;
2;
5;
3.
N
network
is
unable
to
identify
the
minimal
number
of
colors
zwith the smallest
} phase shift. two.
X
X
Thus,
the correct
solution
can
determined
utilizing
the
in and
the therefore
associatedthe
genetic
algorithm)
1 2 With
the initial
conditions
chosen in
the be
simulation
of Fig.optimization
9, thethe
minimum
for
givengoal
graph,
optimization
goal[27].
does
(ϕ)
min properly
Aij network.
cos
(ϕ
ϕ
)10(b)
(12)
i −Fig.
shows
the
evolution
In
the
we
will
transfer
the
crossover
idea
to our
colors, which
the network
is jable
to identify
is three.
As afollowing,
result,
ine, we assign the first vertex,
ϕ 2number of initialized
not
reach
the
global
minimum.
In
contrast
to
the
optimum
0≤j<N
the0≤i<N
application
of the testing
to the phase
ordering
of equation
optimization
goal.strategy
The value
of theshift
optimization
goal
goal
F (ϕ).
Herevertex
ϕ is of
t,ation
if the
second
vertex,
network.
|(8)the
global for
solution
obtained in Fig. 10, as already shown in
already founds {z
the best solution of }
the graph coloring problem
the
th
G(ϕ)
first
decreases
exponentially
and
then
asymptotically
andis ϕnot
thecase,
i
eoscillators
color. This
illustrated
in Fig.
crossoverinitial
should
exchange
the
i isthe
initial conditions =G(ϕ)
chosen in Fig. 9. Note that three is not the As
optimal
solution
Fig.
9
and
in
Fig.
11, 12,
for aa different
state,
the system
to approximately
−6, which
theareglobal
minimum
eected,
of thewhich
ith oscillator.
Aij is converges
of the problem
i.e. the initial conditions
chosenishere
sub-optimal.
is also obvious
elements
of
some
network
identification
vector,
which,
in
our
where we highlighted the new optimization goal G(ϕ). reaches only a local solution [7]. Thereby, the choice of proper
th
column of the adjacency of the optimization problem (12) for this graph. case, corresponds to the phase shift vector, to bypass the local
Fig. 10(b) shows the desired evolution of the phase shifts initial conditions for the dynamic elements of the network
mization goal F (ϕ), we will
However,
is not possible
our network,
since
for a properly initialized network associated to the graph of solutions.
(i.e. capacitors
and this
memristors)
dependsinupon
several system
hase difference, which leads B. Local and global solutions
n to a minimization problem
Depending on the initial state of the system, sometimes the
mization task:
network is unable to identify the minimal number of colors
A cos (ϕ − ϕ )
(12) for the given graph, and therefore the optimization goal does
Fig. 12: I
phase shif
a phase
and can
that ϕi
i ∈ {1,
to recon
can be
transisto
of the
[9]. Thu
(Fig. 13
vertices.
1 chang
other w
when co
of (a) th
the num
seen in
network
the first
in Fig. 1
goal of −
shifts an
5ms a c
local so
in an o
with two
enables
to imple
pairs of
able to
network
i, j ∈ [0
particula
lowest n
1
2 N (N −
we need
After an
determin
1) Ident
In this s
by cros
vertex, w
of color
each ver
after we
itself. M
differen
by appl
vertex o
of one
J. Low Power Electron. Appl. 2022, 12, 22
16 of 30
For each i-value in the set {1, . . . , 6}, the ith cycle of the proposed iterative graph coloring procedure provides the following group classification of the nodes of the undirected
graph of Figure 3(a):
(1)
= {0, 3}, C2
(2)
= {1, 4}, C2
C1
C1
(3)
C1
(4)
C1
(5)
C1
(6)
C1
= {4, 2},
= {2, 5},
= {5, 3},
= {3, 0},
(1)
= {1, 4}, C3
(2)
= {2, 5}, C3
(3)
C2
(4)
C2
(5)
C2
(6)
C2
=
=
=
=
(1)
= {2, 5},
(19)
(2)
= {3, 0},
(20)
(3)
(3)
{5, 3}, C3 = {0}, C4
(4)
{3, 0}, C3 = {1, 4},
(5)
(5)
{0}, C3 = {1, 4}, C4
(6)
{1, 4}, C3 = {2, 5}.
= {1},
(21)
(22)
= {2},
(23)
(24)
In each of cycles 1, 2, 4, and 6, our iterative procedure assigns the vertices of the
graph of Figure 3(a) a minimum number of colors, i.e., 3, which, as expected, is higher
than the chromatic number of the graph itself, i.e., n = 2. In each of these iterations the
same three pairs of vertices are grouped together, thus any number in the set {1, 2, 4, 6}
may be assigned to q, and, as a result, any of Equations (19), (20), (22), and (24) may be
taken as outcome of the graph coloring procedure. The suboptimal solution of the graph
coloring problem is clearly indicated in Figure 8(a), where 3 different colors are used to
mark the arcs of the 3 circular sectors, which in turn host the final destinations of the
2 traces associated to the phase shifts of the oscillator pairs (0, 3), (1, 4), and (2, 5), as well
as in the inset of Figure 8(b), where a distinct color is used to fill each node pair in the set
{(0, 3), (1, 4), (2, 5)}.
Let us now analyse comprehensively the working principles of our graph coloring
strategy for the case of the optimal simulation, which produces Figure 9(a) for the dynamical
behaviour of the memristor currents in the 6 oscillators of the network over the steady-state
time interval t ∈ [9.9, 10] ms. Looking in more detail at the evolution of the memristor
currents over the last part of this time interval, i.e., for t ∈ [9.97, 10] ms, Figure 9(b) shows
the common period18 T of the 6 oscillatory waveforms, and marks the time instants t0 and
t3 , at which im,0 and im,3 cross the threshold value Ith = 0.5 ms, respectively, allowing to
compute the steady-state phase shift ϕ3 (s) between cells 3 and 0, as described in the figure
caption. Computing also the relative phase shifts of the other 5 oscillators of the network
over the common T-long time interval shown in plot (b) of Figure 9(b), the steady-state
phase shift vector ϕ(s) is found to be equal to
ϕ(s)
(s)
(s)
(s)
(s)
(s)
(s)
= [ ϕ0 , ϕ1 , ϕ2 , ϕ3 , ϕ4 , ϕ5 ]T = [0◦ , 180◦ , 5◦ , 195◦ , 11◦ , 182◦ ]T .
(s)
(s)
(25)
(s)
as indicated in Figure 8(c). The resulting phase shift ordering, namely ϕ0 − ϕ2 − ϕ4 −
(s)
(s)
(s)
ϕ1 − ϕ5 − ϕ3 , allows to establish the following vertex ranking:
0 − 2 − 4 − 1 − 5 − 3.
(26)
For each i-value in the set {1, . . . , 6}, the ith cycle of the proposed iterative graph coloring procedure provides the following group classification of the nodes of the undirected
graph of Figure 3(a):
18
in this work the estimation of the common period T of the oscillations developing in a N-cell network, and the
associated group ordering of the phase shifts of the cells 1, . . ., N − 1 relative to the null phase of the reference
cell 0 are carried out every cycle throughout the duration of any simulation.
J. Low Power Electron. Appl. 2022, 12, 22
17 of 30
(1)
= {0, 2, 4}, C2
(2)
= {2, 4, 0}, C2
(3)
= {4, 1}, C2
C1
C1
C1
(4)
C1
(5)
C1
(6)
C1
(1)
= {1, 5, 3},
(27)
(2)
= {1, 5, 3},
(28)
(3)
=
=
=
(3)
= {5, 3}, C3
(4)
{1, 5, 3}, C2 = {0, 2, 4},
(5)
{5, 3, 1}, C2 = {0, 2, 4},
(6)
(6)
{3, 0}, C2 = {2, 4}, C3
= {0, 2},
(29)
(30)
(31)
= {1, 5}.
(32)
In each of cycles 1, 2, 4, and 5, our iterative procedure assigns the vertices of the
graph of Figure 3(a) a minimum number of colors, i.e., 2, which, as expected, coincides
with the chromatic number of the graph itself, i.e., n = 2. In each of these iterations the
same two triplets of vertices are grouped together, thus any number in the set {1, 2, 4, 5}
may be assigned to q, and, as a result, any of Equations (27), (28), (30) and (31) may be
taken as outcome of the graph coloring procedure. The optimal solution of the graph
coloring problem is clearly indicated in Figure 8(c), where 2 different colors are used to
mark the arcs of the 2 circular sectors, which in turn host the final destinations of the 3
traces associated to the phase shifts of the oscillator triplets (0, 2, 4), and (1, 5, 3), as well as
in the inset of Figure 8(d), where a distinct color is used to fill each node triplet in the set
{(0, 2, 4), (1, 5, 3)}.
J. Low Power Electron. Appl. 2022, 12, 22
18 of 30
3
2
1
0
4
3
2
1
0
9.9
9.92
9.94
9.96
9.98
10
4
3
2
1
0
9.97
9.98
9.99
10
Figure 9. (a) Steady-state time evolution of the memristor current im,i of oscillator i ∈ {0, 1, 2, 3, 4, 5}
over the time interval [9.9, 10] ms for the simulation of the balanced network of Figure 3(b) with
compensation for the memristor device-to-device variability and under the optimal initialisation
setting (see also Figure 8(c),(d) for more results obtained from this simulation). The differences in
the peak values of the waveforms originate from the variability in the static and dynamic properties
of the samples, as reproduced by our memristor model. As was already shown in relation to their
relative phases in Figure 8(c), the oscillator triplets (1, 5, 3) and (0, 2, 4) group together, as respectively
indicated over the first and second half of the first observable cycle, where the order of appearance of
the nearby peaks of the memristor currents follows in turn the patterns 1-5-3 and 0-2-4. The same
color coding map, as established in Figure 8(c) and reused in the inset of Figure 8(d), is adopted here
to differentiate between the traces pertaining to distinct oscillators. (b) Zoom-in view of the time
behaviour of each memristor current in the network across the time span [9.97, 10] ms. The common
period T of the oscillatory waveforms is found to be equal to 19.21 µs. As an example, the time instant
t0 (t3 ), at which the memristor current im,0 (im,5 ) of oscillator 0 (3) crosses the threshold Ith = 0.5 mA
in its ascending phase over the first observable cycle, gives 9973.8457µs (9984.2351µs), allowing to
(s)
compute the steady-state phase of oscillator 3 relative to the reference oscillator 0 via ϕ3 = ∆t3 · ω0 ,
with ω0 = 2T·π . Performing a calculation of this kind for each of the remaining 5 oscillators results in
the steady-state phase shift vector ϕ(s) reported in Equation (25).
5. Control Paradigms to Resolve Local Minima-Based Impasse Conditions
In order to overcome the impasse, emerging when a memristive network converges to
a stable oscillatory solution associated to a local minimum of the optimisation goal function
G (ϕ), it is necessary to destabilise the array through an ad-hoc perturbation. We identified
two possible strategies allowing to pull the dynamical system out of a local minimum of
J. Low Power Electron. Appl. 2022, 12, 22
19 of 30
the respective optimisation problem. In many cases, after recovering from the impasse, the
network was found to converge asymptotically toward an oscillatory solution associated
the global minimum of the optimisation goal function. The proposed approaches, referred
to as crossover and pulse destabilisation strategies, are presented in the following two sections.
5.1. Crossover Strategy
One of the strategies, allowing the network to overcome an impasse situation, inspired
from genetic algorithms [45], is based upon the interchange between the connections
of two properly-selected oscillators19 ([31,33]). This corresponds to the exchange of the
associations between the two cells of the network and the corresponding vertices in the
relative graph. In order to select the most appropriate pair of oscillators—let us use indices
i and j to label them—for the crossover, the following two-step procedure is applied.
1.
2.
19
20
21
For each value of k in the set {0, . . . , N − 1}, the vertex k is removed from the original
N-node graph, and the iterative vertex coloring strategy is applied to the resulting
graph of ( N − 1) nodes, using a modified version of the vertex ranking, which is
tabulated beforehand, after a simulation of the oscillatory network, under a generic
sub-optimal initialisation setting, attains the steady state. Specifically, the label of the
vertex k, taken out of the original graph, is removed from the original vertex ranking,
resulting in a new table with N − 1 entries. For each value of k, a N × N matrix,
denoted as A(k) , and obtained from the original adjacency matrix A by setting to 0 all
the elements at row k and at column k, may still be used to define the connectivity of
the respective ( N − 1)-node graph. Coloring the N − 1 vertices of N distinct graphs,
at least one of the N problems will be found to admit the best solution, allowing to
categorise the N − 1 nodes of the relative graph through the lowest number of color
groups. The particular node k, which, extracted out of the original graph, allows the
resulting network to identify the least number of colors according to our iterative
vertex coloring procedure, may then be chosen as first vertex i to involve in the
crossover20 .
Assigning, one at a time, any integer from the set {0, . . . , i − 1, i + 1, . . . , N − 1} to k,
the iterative vertex coloring strategy is then applied to a new vertex ranking, obtained
from the original table by interchanging the positions of vertices i and k. Note that the
original N-node graph, with connectivity defined by the adjacency matrix A, should
be considered in each of the N − 1 applications of the iterative vertex coloring strategy,
since nothing else, except for the correspondence between oscillators and vertices,
is affected in a crossover operation21 . Solving the resulting N − 1 vertex coloring
problems, the solution, assigning the least number of colors to the N nodes of the
original graph, will be determined. It may happen that, on the basis of the proposed
iterative procedure, for two or more values of k, the exchange between the positions
of oscillators k and i in the original vertex ranking results in a common lowest number
of color groups for the N nodes of the original graph. In this case, the choice of
the second oscillator j to involve in the crossover falls for the particular candidate
The application of a crossover to pairs of oscillators implies the necessity to endow the network with reprogrammable connections, e.g. via transistor-based switches, which, however, would add on to the integrated
circuit (IC) overhead in a future hardware implementation of the network.
In fact, it is highly probable that this node mostly prevents the optimisation measure of Equation (16) from
attaining the global minimum, which would provide as solution to the vertex coloring task the chromatic
number of the original N-node graph, as desired. In case, for each of two or more values of k, the application
of our vertex coloring strategy to the respective ( N − 1)-node graph, obtained by removing the vertex k from
the original graph, results in a common lowest number of colors, any of these node i candidates may be finally
considered for the crossover
The interchange between nodes i and k operated on the original vertex ranking is due to the fact that the
application of a crossover between the corresponding oscillators in the network is equivalent to exchanging
their associations to the respective pair of vertices in the original graph. The relative phases, inherent to the
oscillators, maintain the same ordering, as established originally. As a result, the oscillator ranking remains
unaltered, but the mapping from oscillator ranking to vertex ranking is subject to the earlier mentioned
node interchange.
J. Low Power Electron. Appl. 2022, 12, 22
20 of 30
k, whose relative phase ϕk features the largest distance from the relative phase ϕi of
oscillator i in the steady-state phase shift vector ϕ obtained through the simulation
preceding the application of the two-step strategy.
In order to gain a deeper understanding of the proposed two-step procedure, let us
apply it to the vertex ranking 0 − 1 − 4 − 2 − 5 − 3, obtained at steady state from a simulation
of the balanced network of Figure 3(b) (or, equivalently, of Figure 10(b), the corresponding
graph of which is illustrated again in Figure 10(a) for the sake of clarity) with compensation
for the memristor device-to-device variability and under the same sub-optimal initialisation
setting as in the simulation illustrated in Figure 8(a),(b) (see the caption of Figure 11 for
details). According to the first step of the procedure, applying our iterative vertex coloring
strategy to the original vertex ranking, i.e., 0 − 1 − 4 − 2 − 5 − 3, after depriving it of the kth
node label, with the associated 5-node graph obtained from the original one of Figure 3(a)
by breaking all the connections of the kth vertex (k ∈ {0, 1, 2, 3, 4, 5}), the smallest number
of colors, the vertices of the 5-node graph are assigned to, is 2 for each k-value in the
set {0, 1}, and 3 for each k-value in the set {2, 3, 4, 5}. The choice for the i cell for the
crossover may then fall either on reference oscillator 0 or on oscillator 1. Let us choose
the latter cell. In line with the second step of the procedure, exchanging the positions of
labels k and i = 1 in the original vertex ranking 0 − 1 − 4 − 2 − 5 − 3 (k ∈ {0, 2, 3, 4, 5}),
the application of the iterative vertex coloring strategy to the resulting sequence, with
respect to the original 6-node graph of Figure 3(a), results in a minimum number of color
groups equal to 2 for each k-value in the set {0, 2}, to 3 for each k-value in the set {3, 4},
and to 4 for k = 5. Now, given that, in the original steady-state phase shift ordering,
(s)
(s)
(s)
(s)
ϕ2 − ϕ0 = 122◦ > ϕ1 − ϕ0 = 118◦ , oscillator with label k = 2 is selected as cell j for
the crossover. With reference to Figure 10, where plots (a) and (b) show once again the
six-node ring-based graph, and the associated capacitively-coupled array of memristor
oscillators, plot (c), redrawn in a different but equivalent form in plot (d), depicts the novel
coupling arrangement in the network upon the interchange between the connections of
oscillators 1 and 2.
Figure 10. (a) Original 6-node ring-based graph. (b) ((c) or, equivalently, (d)) Coupling arrangement
in the network associated to the graph in (a), before (after) a crossover between cells 1 and 2, which
swaps the correspondence between these cells and the respective nodes in the graph.
The numerical results illustrated in Figure 11, where plots (a), (b), and (c) respectively
show phase dynamics, time evolution of the optimisation goal function, and temporal
trend of the solution of the classification task, respectively, provide evidence for the success
of the circuit implementation of the crossover strategy in pulling the dynamical system
out of the impasse state, allowing its asymptotic convergence to the solution of the graph
coloring problem associated to the global minimum of G (ϕ). These results were obtained
by simulating the balanced network of Figure 10(b), with compensation for the memristor
device-to-device variability, and under the same sub-optimal initialisation setting as in the
simulation illustrated in Figure 8(a),(b). With reference to Figure 11, at the end of the first
part of the simulation, covering the time span t ∈ [0, 5) ms,the optimisation goal function
was found to sit at a local minimum value, specifically −3 (see plot (b)), and the minimum
number of colors, which, on the basis of our iterative vertex coloring procedure, may be
J. Low Power Electron. Appl. 2022,IEEE
12, 22
IEEE TRANSACTIONS
TRANSACTIONS ON
ON CIRCUITS
CIRCUITS AND
AND SYSTEMS—I:
SYSTEMS—I: REGULAR
REGULAR PAPERS
PAPERS
22
11
21 of 30
22
11
11
88
22
assigned to the nodes of the graph in Figure 10(a), is 3 (refer to plot (c)). At t = 5 ms the
33
00
33
00
33
00
connections of oscillators 1 and 2 were interchanged, as shown in plot (c) or, equivalently,
55
55
44
44
in plot (d) of Figure
10. Looking
more at44 Figure 11, 55the dynamics
of the relative
phases of
(b)
(b)
(c)
(c) steady-state pattern,
the cells resume directly(a)(a)after the crossover, approaching
a new stable
whereby
ϕ) is found
to sit
at itstotoglobal
minimum
particularly
−6prior
(see
plot
(b))),
Fig.
Fig. 13:
13:
Illustration
Illustration of
ofG
aa(crossover
crossover
in
in the
the network
network
associated
associated
the
the graph
graph of
of Fig.
Fig. 88 between
betweenlevel,
oscillators
oscillators
11 and
and 2.
2. (a)
(a) Network
Network
prior
to
to the
the
crossover.
crossover.
Oscillator
Oscillator 11
(2)
(2) is
is associated
associated to
to vertex
vertex 11 (2).
(2). (b)
(b) Crossover
Crossover implementation
implementation by
by reconfiguration
reconfiguration of
of the
the connections
connections between
between the
the oscillators.
oscillators. The
The connections
connections from
from oscillator
oscillator
and
the
network
is
able
to
identify
the
chromatic
number
of
the
graph
in
Figure
10(a),
as
22 are
are assigned
assigned to
to oscillator
oscillator 11 and
and vice
vice versa.
versa. (c)
(c) Rearrangement
Rearrangement of
of the
the view
view of
of the
the network
network in
in (b)
(b) to
to enable
enable an
an easier
easier comparison
comparison to
to (a).
(a). Oscillators
Oscillators 11 and
and 22
reverse
reversedetermined
their
their roles
roles in
in the
the network.
network. Oscillator
Oscillator
11 (2)
(2) is
is now
now associated
associated
to
to vertex
vertex
22 (1).
(1). coloring procedure (refer to plot (c)).
through
the proposed
iterative
vertex
90°
90°
10ms
10ms
8ms
8ms
6ms
6ms
4ms
4ms
44
2ms
2ms
0ms
0ms
0°
0°22
11180°
55180°
33
00
22
−2
−2
−3
−3
33
−4
−4
33
11
00
44
55
−5
−5
225°
225°
315°
315°
270°
270°
Number
Number of
of colors
colors
45°
45°
G(ϕ)
G(ϕ)
135°
135°
22
−6
−6
00
(a)
(a)
22
44
66
time
time // ms
ms
(b)
(b)
88
10
10
00
22
44
66
time
time // ms
ms
(c)
(c)
88
10
10
Fig.
Fig. 14:
14: Evolution
Evolution of
of (a)
(a) the
the phase
phase shift.
shift. (b)
(b) the
the optimization
optimization goal
goal G(ϕ)
G(ϕ) and
and (c)
(c) the
the number
number of
of colors
colors for
for the
the graph
graph in
in Fig.
Fig. 8.
8. For
For this
this simulation
simulation the
the same
same
11.
Time
evolution
of the
phases
ofinstant
the oscillators
of the of
balanced
of
initial
initial Figure
state
state as
as the
the
one
one(a)
selected
selected
for
for the
the system
system of
of Fig.
Fig.
99 was
wasrelative
chosen,
chosen, but
but
at
at the
the time
time
instant
tt =
= 5ms
5ms the
the crossover
crossover
of
oscillators
oscillators 11network
and
and 22 was
was implemented.
implemented.
The
The assignment
assignment of
of vertices
vertices to
to color
color groups,
groups, on
on the
the basis
basis of
of the
the final
final phase
phase shifts
shifts of
of the
the respective
respective oscillators,
oscillators, is
is visualized
visualized in
in plot
plot (a)
(a) by
by highlighting
highlighting the
the
Figure
with
compensation
the memristor
and under
the same
perimeter
perimeter
of
of the
the3(b),
left
left (right)
(right)
half
half
circle,
circle, which
which the
the 33for
trajectories,
trajectories,
indicating
indicating the
thedevice-to-device
temporal
temporal trends
trends of
of the
the variability,
phases
phases of
of the
the oscillators
oscillators
corresponding
corresponding
to
to the
the vertices
vertices
0,
0, 2,
2, and
and
4
4
(1,
(1,
3,
3,
and
and
5),
5),
belong
belong
to
to
at
at
steady
steady
state,
state,
with
with
the
the
same
same
color
color
used
used
in
in
plot
plot
(b)
(b)
to
to
fill
fill
the
the
three
three
circles
circles
symbolizing
symbolizing
the
the
corresponding
corresponding
vertices
vertices
of
of
the
the
sub-optimal initialisation setting as in the simulation illustrated in Figure 8(a),(b), for the case where
graph.
graph.
the connections of oscillators 1 and 2 are interchanged at t = 5 ms. Right before the application of
the crossover to the two cells, the network is found to sit on a stable oscillatory solution associated
to a local
minimum
of the on
optimisation
(see also
theof
phase
clusters
emerging
in vertex
additional
additional
simulation
simulation
is
is performed
performed
on
the
the network
network goal
in
in the
thefunction
twotwo- lowest
lowest
number
number
ofthree
colors,
colors,
the
the final
final
choice
choice
goes
goes for
for that
that
vertex
step
step plot
procedure.
procedure.
At
At before
the
the end,
end,the
we
wecrossover
finally
finally select
select
the
the particular
particular
jj candidate,
candidate,
which,
in
in the
the original
original
sorting,
sorting,
features
features
the
the highest
highest
(a) right
procedure).
Despite
the phasewhich,
shift vector,
measured
much
earlier
vertex
vertex i,i, whose
whose removal
removal from
from the
the sorting
sorting results
results in
in the
the least
least phase
phase shift
shift modulus
modulus with
with respect
respect to
to vertex
vertex i.i.
than it was33done in the simulation of Figure 8(a),(b), was found to be slightly different from the one
number
number of
of colors
colors .. Therefore,
Therefore, this
this test
test is
is based
based on
on the
the phase
phase
(
s
)
◦
◦
◦
◦
◦
◦
T
By
By, utilizing
utilizing
this
this, 119
two-step
two-step
strategy,
we
only
only need
need
2N
2N −
−11 tests.
tests.
00 [0 00, 118
in Equation
(17),adjacency
specifically
ϕ A
238
, 359
, 240strategy,
] , thewe
resulting
vertex
A ii
shift
shift reported
vector
vector ϕ
ϕ and
and
on
on the
the modified
modified
adjacency
matrix
matrix
A=
ii.. A
Thus,
Thus, for
for aa network
network with
with ten
ten vertices
vertices we
we reduce
reduce the
the number
number
is
is obtained
obtained
from
from
the
the adjacency
adjacency
matrix
matrix
A
A of
of the
the(18)
graph,
graph,
by
byEvolution of the optimisation goal function over
ordering
remains
defined
by Equation
. (b)
of
of calculations
calculations from
from 45
45 to
to 9.
9. Thus
Thus our
our two-step
two-step method
method saves
saves
setting
setting
all
all its
its elements
elements
on
onnumber
the
the iithth row
row
and
and column
column
to
zero,
zero, through the iterative vertex coloring procedure
time.
(c)
Minimum
of color
groupsto
assigned
aa lot
lot of
of computation
computation time
time for
for larger
larger networks.
networks. As
As an
an example,
example,
which
which represents
represents the
the removal
removal of
of the
the vertex
vertex ii from
from the
the original
original
let us
us time
apply
apply(the
this
this procedure
two-step
two-step method
method
to
to the
the once
example
example
of
of Fig.
Fig. 14.
14.
of Section 4 to the nodes of the graph in Figure 3(a) let
over
is applied
every
graph.
graph.
At
At the
the end
end
of
of the
the first
firstof
part
part
of
ofcurrents
the
the simulation,
simulation,
i.e.
i.e. at
at tt =
= 5ms
5ms
T-long cycle, with the common period T of the oscillatory
waveforms
the
through
the
we
we reach
reach aa steady
steady state
state phase
phase shift
shift as
as shown
shown in
in equation
equation (8).
(8).
2)
2) Identify
Identify the
the second
second vertex
vertex jj 6=
6= ii for
for the
the crossover:
crossover:
memristors, measured over the time interval [4.965,According
4.984] ms,tofound
tostep,
be equal
to 19.24
µs).ofAfter
the
the first
first step,
the
the lowest
lowest number
number of
colors,
colors, which
which
In
In the
the second
second step,
step, the
the other
other vertex
vertex j,
j, involved
involved in
in the
the crossover,
crossover, According to
the crossover
nodes
into
2 groups
(c).
The interchange
between
the couplings
of
the
network
network
is
is able
able to
to identify
identify
for
for each
each
case
case where
where one
one of
of the
the
is
is determined.
determined.
Here,
Here,these
ii is
is fixed
fixed
to
to are
the
the classified
result
result of
of the
the
first
first
step
step the
vertices
vertices
of
of
the
the
associated
associated
graph
graph
removed,
removed,
is
is
calculated.
calculated.
For
For
of
of the
the
strategy.
strategy. To
To
determine
determine
the
the number
number
of
of resolve
colors,
colors, which,
which,
oscillators
1 and
2 was thus
found to
the impasse, allowing the memristive array to approach
the
theofcalculations,
calculations,
we
weidentify
follow
follow the
the
the chromatic
routine
routine of
of section
section III
III using
using
most
mostthe
likely,
likely,
the
the network
network
will
identify
identifytoafter
after
the
the crossover
crossover
optimal
solutionwill
associated
the global
minimum
G (ϕ), and to
number
the steady
steady phase
phase shift
shift vector
vector of
of equation
equation (8)
(8) as
as well
well as
as the
the
between
between vertices
vertices ii and
and j,
j, we
we can
can apply
apply the
the concept
concept described
described the
n = 2IIIofto
the
associated
graph A
(see
also
the
two phase
clusters
emerging
(a)the
at test
the of
end
of 1,
00
modified
adjacency
adjacency
matrix
matrixinA
Aplot
For
the
test
of
vertex
vertex
1, we
we
in
in section
section III
to the
the adjacency
adjacency matrix
matrix A
and
and the
the modified
modified modified
ii.. For
the phase
simulation).
derive the
the following
following modified
modified adjacency
adjacency matrix:
matrix:
crossver
crossver
phase
shift
shift vector
vector ϕ
ϕcc obtained
obtained from
from the
the phase
phase vector
vector derive




ϕ,
ϕ, by
by interchanging
interchanging its
its iithth and
and jjthth elements
elements as
as illustrated
illustrated in
in
00 00 00 00 00 11
Fig.
Fig. 12
12 for
forAs
ii =
=anticipated
11 and
and jj =
= 2:
2: earlier, the circuit implementation of the crossover
crucially
re
00 00strategy


00 00 00 00









quires the availability
of
reprogrammable
connections
among
the
oscillators
of
the
network.
0
0
0
0
0
0
1
1
0
0
0
0

ϕ
ϕjj ifif kk =
= ii


 ..


A
A0011 =
=
(14)

00 00Furthermore,
the cir- (14)
11 00 11 00
This inevitably
of the hardware platform.




ϕ
ϕcckk =
=increases
(13)
(13)
ϕ
ϕii ifif kk the
=
= jj area overhead






0
0
0
0
0
0
1
1
0
0
1
1

cuitry necessary
to
the..network with reprogrammable connectivity, may introduce
ϕ
ϕkkendow
otherwise
otherwise
11 00 00 00 11 00
some mismatch between the capacitive loads of the oscillators, which may represent an
If,
If, two
two or
or more
more vertex
vertex jj candidates,
candidates, after
after the
the crossover
crossover with
with The
The modified
modified
network
network
may
may group
group
the
the remaining
remaining
five
vertices
obstacle
toward
thenetwork
convergence
of thethe
phase
dynamics
of the
memristive
array
toward five
the vertices
vertex
vertex i,i, lead
lead to
to aa modified
modified network
able
able to
to identify
identify the
same
same into
into two
two color
color sets
sets i.e.
i.e. CC11 =
= {0,
{0, 4,
4, 2},
2}, and
and CC22 =
= {5,
{5, 3}.
3}. The
The
pattern corresponding to the global minimum ofidentification
the optimisation
problem.
For
this reason,
identification of
of two
two colors
colors is
is also
also achieved
achieved by
by the
the modified
modified
another
strategy
for
pulling
the
dynamical
system
out
of
a
local
minimum,
which
is original
more
network
network
obtained
obtained
by
by
taking
taking
vertex
vertex
0
0
out
out
of
of the
the
original graph,
graph,
modified
modified network
network identifies
identifies the
the same
same lowest
lowest number
number of
of colors,
colors, any
any of
of these
these
vertices
vertices
may
may be
be selected
selected
first
first vertex
vertex
for
for the
the crossover.
crossover.
while
while
in
each
each
of
of the
the
other
other cases
cases the
the resulting
resulting modified
modified network
network
amenable
toasascircuit
implementation,
is presented
in in
the
next
section.
33If,
If, as
as aa result
result of
of the
the removal
removal of
of any
any of
of two
two or
or more
more vertices,
vertices, the
the resulting
resulting
5.2. Pulse Destabilisation Strategy
Inspired from simulated annealing algorithms [46] as well from our latest approach—
referred to as Kick-Fly-Catch paradigm—for humanoid robot motion control [47,48], the
proposed strategy ([31,33]) is based upon supplying energy to the system, while it sits in
an impasse state. Particularly, a pulse is applied to an ad-hoc oscillator by offsetting the
value VS of its DC voltage source by an appropriate amount ∆VS for a given time interval
of length Tp . Consequently, the relative phase of the oscillator, sitting at the steady-state
J. Low Power Electron. Appl. 2022, 12, 22
22 of 30
level ϕ(s) previous to the stimulation, experiences a sudden shift by ∆ϕ, which is found to
depend upon amplitude ∆VS and length Tp of the voltage pulse. Taking inspiration from
the research study presented in [49], extensive numerical simulations revealed that, for a
given pulse width22 Tp , there is an approximately-linear relation between the pulse height
∆VS and the sudden shift ∆ϕ, that the relative phase of the oscillator experiences upon
destabilisation, i.e.,
∆VS ≈
V0 · ∆ϕ
,
180◦
(33)
where V0 was numerically set to −0.23 V for the networks analysed in this research study23 .
Provided the right choice is made regarding the cell to perturb, and the proper amplitude
and width are assigned to the voltage pulse stimulus, the resulting resumption of the phase
dynamics of the oscillators should ideally allow the optimisation goal function to move out
of the local minimum, converging asymptotically toward its global minimum. A two-step
procedure, presented below, is proposed here to determine the most suitable cell to perturb
and the most appropriate pulse amplitude.
1.
2.
The most suitable oscillator i ∈ {0, . . . , N − 1} to target in the pulse destabilisation action is determined in the same way as was done for the selection of the cell
i ∈ {0, . . . , N − 1} to involve in the crossover process (see the first step in the procedure aimed to choose the right cell pair (i, j) to involve in the coupling interchange strategy).
The second step is aimed to determine the appropriate shift ∆ϕ to be added to the
(s)
steady-state relative phase ϕi of oscillator i for pulling the network out of the local
minimum state, facilitating its convergence to an oscillatory solution, which would
ideally correspond to the least number of color groups for the N vertices of the associated graph. To accomplish this task, for each value of k within the set {1, . . . , M − 1},
◦
with M a predefined positive integer, the offset ∆ϕk = k · 360
M is added to the phase
(s)
shift ϕi of cell i in the steady-state relative phase shift vector ϕ(s) recorded before
the application of the pulse destabilisation process, and the iterative graph coloring
procedure is applied to the resulting vertex ranking for the original graph. The choice
for the most appropriate offset ∆ϕ, within the specified set of k-dependent uniformlyspaced values, goes for the ∆ϕk -candidate, which, according to our graph coloring
strategy, allows the network to classify the nodes of the associated graph in the lowest
number of color groups. If, for two or more k-values, the application of the iterative
graph coloring procedure to the vertex ranking, resulting from the phase shift order(s)
ing, obtained by adding up the relevant offset ∆ϕk to the phase shift ϕi of oscillator
i in the steady-state relative phase shift vector ϕ(s) , leads to the identification of the
same lowest number of colors, the selection goes for the ∆ϕk -candidate featuring the
largest modulus. Finally, the pulse amplitude of the Tp -long stimulus to be applied to
oscillator i is obtained from Equation (33).
In order to gain more insights into the mechanisms underlying this two-step procedure,
let us apply it to the vertex ranking 0 − 1 − 4 − 2 − 5 − 3, obtained at steady state from a
simulation of the balanced network of Figure 3(b) with compensation for the memristor
device-to-device variability and under the same sub-optimal initialisation setting as in the
simulation illustrated in Figure 8(a),(b) (see the caption of Figure 12 for details). Given
that the first step of the procedure is identical to the first step of the algorithm allowing to
22
23
In this work Tp was set to twice the common graph-dependent period T of the oscillatory waveforms of
the capacitor voltages and of the memristor currents in the network before the application of the pulse
destabilisation paradigm.
We acknowledge, however, that the most suitable formula, expressing the relationship between the amplitude
∆VS of a destabilising pulse of fixed width Tp and the resulting sudden shift ∆ϕ in the phase of the perturbed
oscillator, may depend upon network properties and parameters. A deeper study, aimed to optimise the shape
of the destabilisation stimulus, will be carried out in the future.
J. Low Power Electron. Appl. 2022, 12, 22
23 of 30
select the optimal cell pair (i, j) to involve in the crossover strategy, retrieving the results
presented in the previous section, either oscillator from the label set i ∈ {0, 1} may be
chosen as target of the pulse destabilisation action. Let us go for the latter one. Following
the guidelines established for the second step of the procedure, setting M to 4, for each kvalue in the set {1, 2, 3}, colors are assigned to the nodes of the original graph of Figure 3(a)
by applying the proposed iterative methodology to the kth vertex ranking variant derived
from the phase shift vector ϕ(s) = [0◦ , 118◦ , 238◦ , 359◦ , 119◦ , 240◦ ]T recorded right before
the time instant24 t = 5 ms, at which the pulse perturbation action is commenced by adding
(s)
up the kth value of ∆ϕk in the set {90◦ , 180◦ , 270◦ } to the relative phase ϕ1 of oscillator
1. Analysing the kth vertex ranking within the set {0 − 4 − 1 − 2 − 5 − 3, 0 − 4 − 2 − 5 −
1 − 3, 0 − 1 − 4 − 2 − 5 − 3} (k ∈ {1, 2, 3}), according to our node coloring paradigm the
network identifies a minimum number of color groups equal to the kth number in the set
{3, 2, 3}. Setting k to 2 in the formula for ∆ϕk , from Equation (33) the amplitude ∆VS of
the
pulseONapplied
toAND
oscillator
1 REGULAR
at
t =PAPERS
5PAPERS
ms for a Tp = 38.48 µs-long time interval is set to
IEEE
IEEE
TRANSACTIONS
TRANSACTIONS
ON
ONCIRCUITS
CIRCUITS
AND
SYSTEMS—I:
SYSTEMS—I:
REGULAR
PAPERS
IEEE TRANSACTIONS
CIRCUITS
AND
SYSTEMS—I:
REGULAR
−0.23 V (see the caption of Figure 12 for details).
10 101
90°
90°90°
10ms
10ms
10ms
8ms
8ms
8ms
6ms
6ms
6ms
4ms
4ms
4ms
2 22
2ms
2ms
2ms
0ms
0ms
0ms
44
0° 40°0°
0 00
2 22
1 11
−2−2−2
−3−3−3
−4−4−4
3 33
4 44
0 00
5 55
−5−5−5
225°
225°
225°
315°
315°
315°
270°
270°
270°
(a)(a)(a)
−6−6−6
0 0 0 2 2 2 4 4 4 6 6 6 8 8 8 10 1010
time
time
/ ms
/ ms
time
/ ms
(b)
(b)(b)
3 33
Number of colors
Number of colors
Number of colors
5180°
5180°
5180°
1 11
3 33
45°
45°45°
G(ϕ)
G(ϕ)
G(ϕ)
135°
135°
135°
2 22
0 0 0 2 2 2 4 4 4 6 6 6 8 8 8 10 1010
time
/ ms
timetime
/ ms/ ms
(c) (c)(c)
Fig.
15:15:
Evolution
of of
(a)
the
phase
shift.
(b)(b)
the
optimization
goal
G(ϕ)
andand
(c)
the
number
of colors
for for
the
graph
in Fig.
8.
For
this
simulation
the the
same
Fig.
Fig.
15:
Evolution
Evolution
of(a)
(a)the
thephase
phase
shift.
shift.
(b)the
theoptimization
optimization
goal
goal
G(ϕ)
G(ϕ)
and(c)
(c)the
thenumber
number
ofofcolors
colors
forthe
thegraph
graph
ininFig.
Fig.
8.8.For
Forthis
thissimulation
simulation
thesame
sam
Figure
12.
(a)
Phase
dynamics
of
the
balanced
network
of
with
for
thein ainphase
initial
state
as as
the
one
selected
forfor
the
system
of of
Fig.
9 was
chosen,
butbut
at
time
instant
tFigure
=t 5ms
a3(b),
pulse
was
applied
to oscillator
1, resulting
initial
initial
state
state
asthe
the
one
one
selected
selected
forthe
thesystem
system
ofFig.
Fig.
9 9was
was
chosen,
chosen,
butthe
at
atthe
thetime
time
instant
instant
t==5ms
5ms
a apulse
pulse
was
wascompensation
applied
applied
totooscillator
oscillator
1,
1,resulting
resulting
ina aphase
phas
shift
ofofapproximately
180°.
The
assignment
of of
vertices
to to
color
groups,
on on
the
basis
of the
final
phase
shifts
of the
respective
oscillators,
is visualized
in ini
shift
shift
ofapproximately
approximately
180°.
180°.
The
The
assignment
assignment
ofvertices
vertices
tocolor
color
groups,
groups,
onthe
thebasis
basis
ofofthe
thefinal
final
phase
phase
shifts
shifts
ofofthe
therespective
respective
oscillators,
oscillators,
isisvisualized
visualized
memristor
device-to-device
variability,
and
under
the
same
sub-optimal
setting
inthe
plot
(a)(a)(a)
by
thethe
perimeter
of of
the
left
(right)
halfhalf
circle,
which
thethe
3the
trajectories,
indicating
the the
temporal
trends
of the
phases
ofas
the
oscillators
plot
plot
byhighlighting
byhighlighting
highlighting
the
perimeter
perimeter
ofthe
theleft
left
(right)
(right)
half
circle,
circle,
which
which
3 3trajectories,
trajectories,
indicating
indicating
theinitialisation
temporal
temporal
trends
trends
ofof
the
thephases
phases
ofof
theoscillators
oscillator
corresponding
to to
the
vertices
0,
2,0,2,
and
4 (1,
3,
and
5),5),
belong
to at
state,
with
thewhere
same
color
used
in plot
(b)
to(b)
fill
the
three
circles
symbolizing
the the
corresponding
corresponding
tothe
the
vertices
vertices
0,
2,and
and
4 4(1,
(1,
3,
and
and
5),belong
belong
totosteady
atatsteady
steady
state,
state,
with
with
the
thesame
same
color
color
used
used
ininplot
plot
totothe
fill
fillthe
thethree
three
circles
circles
symbolizing
symbolizing
th
the simulation
illustrated
in3,
Figure
8(a)-(b),
for
the
case
the
voltage
VS(b)
of
DC
source
in
corresponding
vertices
of of
the
graph.
corresponding
corresponding
vertices
vertices
ofthe
thegraph.
graph.
oscillator 1 is offset by ∆Vs = −0.23 V from the time instant t = 5 ms for a temporal window of
duration Tp = 38.48 µs (the common period T of the oscillatory waveforms of the currents through
the
memristors,
measured
over
the
interval
[4.965,
4.984] ms,
was now
found
tofeaturing
be equal
to phase
19.24
µs).
destabilized
network
featuring
a new
phase
shift
ordering,
likely,
would
bebebe
identifiable
byby
thethe
network
in time
case
the
phase
destabilized
destabilized
network
network
now
now
featuring
a anew
new
phase
shift
shift
ordering,
ordering
likely,
likely,
would
would
identifiable
identifiable
by
the
network
network
in
incase
case
the
the
phase
phase
namely
0;
4;
2;
5;
1;
3
(0;
4;
1;
2;
5;
3
or
0;
1;
4;1;4;
2;4;2;
5;2;5;5
shift
ϕAs
of
oscillator
i
experienced
a
sudden
translation
by
namely
namely
0;
0;
4;
4;
2;
2;
5;
5;
1;
1;
3
3
(0;
(0;
4;
4;
1;
1;
2;
2;
5;
5;
3
3
or
or
0;
0;
1;
shift
shift
of
of
oscillator
oscillator
i
i
experienced
experienced
a
a
sudden
sudden
translation
translation
by
by
iϕϕ
discussed
earlier,
right
before
the
application
of
the
pulse
to
cell
1,
which
triggers
a
sudden
shift
ii
3, 3,
respectively)
is is
able
to to
group
thethe
vertices
intointo
twotwo
(three)
any
offset
∆ϕ
ainaset
ofofM
−−
1−1uniformly-spaced
values
3,
respectively)
respectively)
is
able
able
to
group
group
the
vertices
vertices
into
two
(three)
(three
any
any
offset
offset
∆ϕ
∆ϕ
in
a
set
set
of
M
M
1
uniformly-spaced
uniformly-spaced
values
values
◦
j jin
j
in its relative
phase by approximately 180 , the network is found to sit on a stable oscillatory solution
◦ ◦◦
AsAs
aAsresult
, in, ,in
order
to to
change
thethe
phase
shift
ϕ1 ϕof
(∆ϕ
j=·j360
/M
, with
j=
1,
. 1,
. . ., .M
1).
Thereby
wewe
cancan
colors.
colors.
a aresult
result
inorder
order
tochange
change
thephase
phase
shift
shift
ϕ
(∆ϕ
(∆ϕ
j· 360
· 360
/M
/M
, ,with
with
j j=
=1,
,. M
,−
M−
−1).
1).
Thereby
Thereby
we
cancolors.
j j=j=
1 1ofo
associated
to
a local minimum
optimisation
goal function
(see =
also
the
◦ ◦three
reapply
the
concepts
exposed
in insection
IIIof
tothe
adjacency
1 by
∆ϕ
180
, at,◦ ,at
t at=t tphase
5ms
a clusters
Ta-long
voltage
2∆ϕ
reapply
reapply
the
the
concepts
concepts
exposed
exposed
insection
section
IIIIII
tothe
tothe
the
adjacency
adjacencyoscillator
oscillator
oscillator
11by
by∆ϕ
=180
180
=
=5ms
5ms
aTT-long
-long
voltage
voltag
2 2=
p p p destabilisation action). Despite the phase shift vector,
emerging
plot
(a)phase
right
before
theϕpulse
of of
amplitude
∆V∆V
= −0.23V
, as, ,as
follows
from
equation
matrix
AAA
and
totoatoin
shift
vector
matrix
matrix
and
and
aperturbed
aperturbed
perturbed
phase
phase
shift
shift
vector
vector
ϕ,ϕobtained
, ,obtained
obtainedpulse
pulse
pulse
ofamplitude
amplitude
∆V
==−0.23V
−0.23V
asfollows
follows
from
from
equation
equation
from
the
phase
vector
ϕϕ
by
translating
phase
shift
ϕiϕsimulation
of
(15),
is
added
to
the
voltage
V
of
the
DC
source
of
oscillator
1 11
S
from
from
the
the
phase
phase
vector
vector
ϕbyby
translating
translating
the
the
phase
phase
shift
shift
ϕ
of
of
(15),
(15),
is
is
added
added
to
to
the
the
voltage
voltage
V
V
of
of
the
the
DC
DC
source
source
of
of
oscillator
oscillator
measured
much
earlier
than itthe
was
done
in
the
of
Figure
8(a),(b),
was
found
to
be
slightly
ii
SS
itself,
resulting
the
of◦the
optimization
oscillator
i by
a aphase
offset
∆ϕ
◦ , 238◦approach
◦ , 119
◦ ]optimization
Toptimization
j : j :j :
oscillator
oscillator
i ibyby
aphase
phase
offset
offset
∆ϕ
∆ϕ
itself,
itself,
resulting
resulting
the
asymptotic
asymptotic
approach
approach
of
the
the
different
from
the
one
reported
in Equation (17), specifically
ϕ(s) in
= in
[0in◦the
,asymptotic
118
, 359
, of
240
,
(((
goal
G(ϕ)
to to
the
global
solution,
as shown
in Fig.
15.15.
Further
goal
goal
G(ϕ)
G(ϕ)
tothe
theglobal
global
solution,
solution,
asasshown
shown
ininFig.
Fig.
15.Further
Furthe
the resulting
vertex
ordering
(18)
Timegraphs
evolution
of
optimisation
ϕiϕ+
∆ϕ∆ϕ
kremains
i=i i defined by Equation
j j jif if
ϕ
if=
kk=
examples
of. (b)
larger
willwill
be the
discussed
in in
section
V. V.V
i i++∆ϕ
p=
p
examples
examples
of
of
larger
larger
graphs
graphs
will
be
be
discussed
discussed
in
section
section
ϕpϕkϕ
(16)
(16)
(16) assigned through the iterative vertex coloring
k k==
goal function.
(c)
number
ϕkϕMinimum
otherwise.
The
benefits
of of
this
concept
areare
thethe
ease
andand
unconstrained
ϕ
otherwise.
otherwise.of color groups
The
The
benefits
benefits
ofthis
this
concept
concept
are
the
ease
ease
and
unconstrained
unconstrained
kk
nature
of of
the
selection
process
of of
an
appropriate
phase
offset
procedure
of Section
4 tocandidates,
the nodes after
of theaddition
graph in
Figure
3(a)
over
time
(the
procedure
is
once
nature
nature
ofthe
the
selection
selection
process
process
ofan
anapplied
appropriate
appropriate
phase
phase
offset
offse
If,If,If,
two
orormore
phase
offset
∆ϕ
to to
j j jcandidates,
two
two
ormore
more
phase
phase
offset
offset
∆ϕ
∆ϕ
candidates,after
afteraddition
addition
toby which to translate the phase shift a given vertex in order
by
by
which
which
to
to
translate
translate
the
the
phase
phase
shift
shift
a
a
given
given
vertex
vertex
in
in
order
orde
every
T-long
cycle).
After
the
pulse
destabilisation
2
colors
are
assigned
to
these
nodes.
The
pulsethe
phase
shift
ofofvertex
i, i,
lead
to to
atomodified
network
with
thethe
the
the
phase
phase
shift
shift
ofvertex
vertex
i,lead
lead
a amodified
modified
network
network
with
with
theto allow the network to reach the global solution and and its
to
toallow
allowthe
the
thenetwork
networktoto
reach
reachthe
thethe
global
global
solution
solutionand
andand
anditsit
same
lowest
number
of
colors,
the
choice
falls
on
the
candidate
based
perturbation
of
oscillator
1
was
thus
found
to
resolve
impasse,
allowing
memristive
same
samelowest
lowestnumber
numberofofcolors,
colors,the
thechoice
choicefalls
fallsononthe
thecandidate
candidatesimple circuit implementation. But the impact of the amplitude
simple
simplecircuit
circuitimplementation.
implementation.But
Butthe
theimpact
impactofofthe
theamplitude
amplitud
with
the
largest
modulus.
According
to
equation
(15)
thethe
with
with
the
the
largest
largest
modulus.
modulus.
According
According
toto
equation
equation
(15)
(15)
the
array
to approach
the
optimal
solution
associated
toor
the
minimum
(ϕnetwork
)network
, andperformance
toperformance
identifydepends
thedepends
width
of of
the
pulse
onof
the
network
on on
or
orglobal
width
width
ofthe
thepulse
pulse
on
onG
the
the
performance
depends
on
corresponding
amplitude
∆V
of
the
T
-long
pulse
to
be
added
corresponding
corresponding
amplitude
amplitude
∆V
∆V
ofof
the
theTthe
T-long
-long
pulse
pulsetotobe
beadded
added
thethe
system
parameters,
thus
thethe
determination
of plot
anofan
accurate
chromatic
number
n
=
2
of
associated
graph
(see
also
the
two
phase
clusters
emerging
in
the
system
system
parameters,
parameters,
thus
thus
the
determination
determination
of
an
accurate
accurat
upup
to
the
nominal
value
V
of
the
DC
voltage
source
of
S
uptotothe
thenominal
nominalvalue
valueVV
theDC
DCvoltage
voltagesource
sourceofofestimation of the relation between the pulse parameters and
SS ofof the
estimation
estimation
therelation
relationthe
between
between
thepulse
pulseparameters
parametersand
and
(a) iatis
enddetermined.
of
the simulation). Despite, at the time
instant
t =ofof
5 the
ms,
when
pulsethe
perturbation
oscillator
oscillator
oscillator
i ithe
isfinally
isfinally
finally
determined.
determined.
thethe
resulting
translation
in in
the
phase
shift
of of
the
perturbed
the
resulting
resulting
translation
translation
inthe
the
phase
phase
shift
shift
ofthe
theperturbed
perturbed
commences,
G
(
ϕ
)
undergoes
a
sudden
increase
from
the
local
minimum
value
of
−
3,
it
descends
such
as as
the
oneone
proposed
in in
(15)
forfor
the
graphs
Let
ususus
apply
this
two-step
method
to tothe
example
of of
Fig.
15.15.
oscillator,
oscillator,
such
such
asthe
the
one
proposed
proposed
in(15)
(15)
forthe
thegraphs
graph
Let
Let
apply
apply
this
this
two-step
two-step
method
method
tothe
the
example
example
ofFig.
Fig.
15.oscillator,
steeply
straight
away,
decreasing
monotonically
toward
the
global
minimum
value
of
−
6
thereafter.
analyzed
in
this
study,
might
turn
out
to
be
a
difficult
tasktask
AtAt
the
end
of
the
first
half
of
the
simulation
i.e
at
t
=
5ms
the
analyzedininthis
thisstudy,
study,might
mightturn
turnout
outtotobebea adifficult
difficult
task
Atthe
theend
endofofthe
thefirst
firsthalf
halfofofthe
thesimulation
simulationi.ei.eatatt t==5ms
5msthe
the analyzed
more
complex
networks.
Also
thethe
implementation
of this
steady
state
phase
shift
vector
is isexpressed
byby
equation
(8),(8),
for
more
more
complex
complex
networks.
networks.
Also
Also
theimplementation
implementation
ofofthis
thi
steady
steady
state
state
phase
phase
shift
shift
vector
vector
isexpressed
expressed
by
equation
equation
(8),forfor
cancan
notnot
guarantee
immaculate
determination
of ofo
analogously
asasas
inin
thethe
crossover
simulation
of ofFig.
14.
The
The
numerical
results
illustrated
in14.
Figure
12,
where
plots
(a),
(b),anand
(c)
respectively
algorithm
algorithm
can
not
guarantee
guarantee
an
animmaculate
immaculate
determination
determination
analogously
analogously
in
the
crossover
crossover
simulation
simulation
ofFig.
Fig.
14.
The
Thealgorithm
thethe
global
solution,
butbut
it enables
the
system
totrend
overcome
a a
application
ofof
the
first
step
ofofthe
strategy
provides
analogous
show
phase
dynamics,
time
evolution
of
the optimisation
goal
function,
and
temporal
the
global
global
solution,
solution,
but
ititenables
enables
the
thesystem
system
totoovercome
overcome
application
application
ofthe
the
first
first
step
step
ofthe
the
strategy
strategy
provides
provides
analogous
analogous
percentage
of of
local
solutions.
results
asasas
obtained
ininsection
IV-C.
Vertices
0 0and
1 1play
a a alarge
large
large
percentage
percentage
oflocal
local
solutions.
solutions.
results
results
obtained
obtained
insection
section
IV-C.
IV-C.
Vertices
Vertices
0and
and
1play
play
major
rolerole
ininin
preventing
thethe
network
from
determining
thethe
major
major
preventing
preventing
the
network
network
from
from
determining
determining
the
24 role
In
order to
presentoscillator
a fair comparison
between
the beneficial effects of the crossover
and pulse destabilisation
chromatic
number.
Selecting
1 as
target
of of
the
pulse
V. V.
R
chromatic
chromatic
number.
number.
Selecting
Selectingoscillator
oscillator
11as
astarget
target
ofthe
the
pulse
pulse
V.ESULTS
RRESULTS
ESULTS
control
paradigms,
wecandidates
ensured that
the simulations
in Figures 11 and 12 provided identical results for
destabilization
process,
the
offset
are
chosen
freely
In
this
section
the
results
for
the
vertex
coloring
of selected
destabilization
destabilization
process,
process,
the
the
offset
offset
candidates
candidates
are
are
chosen
chosen
freely
freely
◦initialisation
◦
InInassigning
this
thissection
section
the
theresults
results
for
forthe
thevertex
coloring
coloring
selected
t 3∈ values,
[0, 5) msi.e.
by ∆ϕ
choosing
the◦ same
setting, and
a common
random
set
ofvertex
α-values
to theofofselected
from
a aset
{90
, ◦180
, ◦270
},◦ },
◦withj ∈
j ∈∈
from the 2nd DIMACS implementation challenge will
from
from
aset
set33values,
values,i.e.
i.e.∆ϕ
∆ϕ
{90
,◦180
, 180
,◦270
, 270
},withj
withj∈∈graphs
j j ∈{90
memristors.
graphs
graphs
from
from
the
the
2nd
2nd
DIMACS
DIMACS
implementation
implementation
challenge
will
wil
{1, 2, 3} (M = 4). Consequently, three perturbed phase shift be presented. In Table II, the selected graphs, thechallenge
number
{1,
{1,2,2,3}3}(M
(M==4).
4).Consequently,
Consequently,three
threeperturbed
perturbedphase
phaseshift
◦shift be
be
presented.
presented.
In
In
Table
Table
II,
II,
the
the
selected
selected
graphs,
graphs,
the
the
number
numbe
vectors may be computed. For example, for ∆ϕj = 180 , ◦ ◦ of their vertices, and their chromatic number, which is the
vectors
vectorsmay
maybebecomputed.
computed.For
Forexample,
example,for
for∆ϕ
∆ϕ
180, , ofoftheir
j j ==180
theirvertices,
vertices,and
andtheir
theirchromatic
chromaticnumber,
number,which
whichisisthe
th
from equation (8) we obtain
minimal number of different colors by which their vertices
from
fromequation
equation(8)
(8)we
weobtain
obtain
minimal
minimalnumber
numberofofdifferent
differentcolors
colorsbybywhich
whichtheir
theirvertices
vertice
>
be colored, are given on the left side. On the right side of
.> (17) may
ϕp p=p 0◦ ◦ ◦298◦ ◦ ◦240◦ ◦ ◦358◦ ◦ ◦120◦ ◦ ◦242◦ ◦ ◦>
maybebecolored,
colored,are
aregiven
givenononthe
theleft
leftside.
side.On
Onthe
theright
rightside
sideofo
298 240
240 358
358 120
120 242
242
ϕϕ == 00 298
. . (17)
(17)themay
same table the performance results of oscillatory networks
the
thesame
sametable
tablethe
theperformance
performanceresults
resultsofofoscillatory
oscillatorynetworks
network
For each perturbed phase shift vectors, adopting the testing and of the Brelaz heuristic in solving vertex coloring problems
For
Foreach
eachperturbed
perturbedphase
phaseshift
shiftvectors,
vectors,adopting
adoptingthe
thetesting
testing and
andofofthe
theBrelaz
Brelazheuristic
heuristicininsolving
solvingvertex
vertexcoloring
coloringproblems
problem
J. Low Power Electron. Appl. 2022, 12, 22
24 of 30
of the solution of the classification task, respectively, provide evidence for the success of the
circuit implementation of the pulse destabilisation strategy in pulling the dynamical system
out of the impasse state, allowing its asymptotic convergence to the solution of the graph
coloring problem associated to the global minimum of G (ϕ). These results were obtained
by simulating the balanced network of Figure 10(b), with compensation for the memristor
device-to-device variability, and under the same sub-optimal initialisation setting as in
the simulation illustrated in Figure 8(a),(b). With reference to Figure 12, at the end of
this first part of the simulation, covering the time interval expressed as t ∈ [0, 5) ms, the
optimisation goal function was found to sit at a local minimum value, specifically −3 (see
plot (b)), and the minimum number of colors, which, on the basis of our iterative vertex
coloring procedure, may be assigned to the nodes of the graph in Figure 3(a), is 3 (refer
to plot (c)). From the time instant t = 5 ms and for a Tp = 38.48 µs-long time interval,
the offset ∆VS = −0.23 V is added up to the nominal value VS of the DC voltage source
in oscillator 1. As may be evinced by inspecting Figure 11, the relative phase of cell 1
undergoes a sudden shift by approximately 180◦ , and, thereafter, the phase dynamics of the
network evolve toward a new stable steady-state pattern, whereby G (ϕ) is found to sit at
its global minimum level, particularly −6 (see plot (b))), and the network is able to identify
the chromatic number of the graph in Figure 10(a), as determined through the proposed
iterative vertex coloring procedure (refer to plot (c)).
5.3. Discussion
Since a single crossover or pulse destabilisation manoeuvre may be unable to pull a
more complex memristive network out of an impasse situation, or to reach the solution
associated to the global minimum of the optimisation goal function, in case the dynamical
system receives enough energy to move out of the solution associated to a local minimum
of the optimisation goal function25 , a good approach to address this issue would be to
reapply either of the two proposed strategies periodically, interchanging the connections of
two distinct appropriate oscillators or applying a suitable destabilising pulse to a different
ad-hoc oscillator at regular time intervals26 .
Let us provide a proof of principle of the proposed approach, focusing on the pulse
destabilisation control strategy. Similar results were obtained through the periodic application of the crossover paradigm. With reference to Figure 13, plot (a) shows the time
evolution of the phase shifts of the oscillators with labels running from 1 to 24 with respect
to the reference cell 0 in a capacitively-coupled network of NbOx memristor oscillators
implementing a N = 25-node undirected graph known as queen5_5 [50]—see also the inset
in plot (b)—in case compensation for the mismatch between the capacitive loads of the
oscillators and for the memristor device-to-device variability is set in place, and for the case
where, periodically, on the basis of the two-step strategy introduced in Section 5.2, a distinct
oscillator is perturbed by means of a Tp -long pulse of appropriate height ∆Vs . As shown in
plot (b) the optimisation goal function evolves progressively through various local minima
before attaining the global minimum value, which is associated to the identification of the
chromatic number of the queen5_5 graph, i.e., n = 5, as demonstrated in plot (c). After a
38 ms-long transient time interval, the network exhibits a robust oscillatory solution, given
that the subsequent application of destabilisation pulse stimuli to the oscillators does no
longer affect the phase dynamics.
25
26
In some cases, after overcoming the impasse situation, the dynamical system could approach a new oscillatory
solution associated to another local minimum of G (ϕ).
The time separation Tint between consecutive applications of the crossover or pulse destabilisation strategy
is set to 2 ms in the simulations discussed in this section. Furthermore, in this work the estimation of the
common period T of the oscillations appearing in a N-cell network, and the associated group ordering of the
phase shifts of the cells 1, . . ., N − 1 relative to the null phase of the reference cell 0 are carried out every cycle
throughout the duration of any simulation. Moreover, in the first (latter) control strategy, the application of a
pulse to the same oscillator (a crossover involving either oscillator from the same pair) is not allowed until at
least 5 iterations of the control strategy have elapsed first. As a result, each pulse destabilisation (crossover)
manoeuvre targets a different oscillator (involves a different pair of oscillators).
J. Low Power Electron. Appl. 2022, 12, 22
25 of 30
IEEE
IEEE
IEEE
TRANSACTIONS
TRANSACTIONS
TRANSACTIONS
ONON
ON
CIRCUITS
CIRCUITS
CIRCUITS
AND
AND
AND
SYSTEMS—I:
SYSTEMS—I:
SYSTEMS—I:
REGULAR
REGULAR
REGULAR
PAPERS
PAPERS
PAPERS
12 12
12
90°90°
90°
100ms
100ms
100ms
80ms
80ms
80ms
60ms
60ms
60ms
40ms
40ms
40ms
20ms
20ms
20ms
0ms
0ms
0ms
0° 0°
0°
40 40
40
14 14
14
20 20
20
12 12
12
Number of colors
Number of colors
180°
180°
180°
45°45°
45°
10 10
10
G(ϕ)
G(ϕ)
135°
135°
135°
0 00
8 88
−20−20
−20
225°
225°
225°
315°
315°
315°
6 66
−40−40
−40
0 00
270°
270°
270°
(a)(a)
(a)
20 20
20 40 40
40 60 60
60 80 80
80 100100
100
time
time
time
/ ms
// ms
ms
(b)(b)
(b)
0 00
20 20
20 40 40
40 60 60
60 80 80
80 100100
100
time
time
time
/ ms
// ms
ms
(c)(c)
(c)
Fig.Fig.
Fig.
16:16:
16:
Evolution
Evolution
Evolution
of of
(a)
of (a)
(a)
thethe
the
phase
phase
phase
shift.
shift.
shift.
(b)(b)
(b)
thethe
the
optimization
optimization
optimization
goal
goal
goal
G(ϕ)
G(ϕ)
G(ϕ)
andand
and
(c)(c)
(c)
thethe
the
number
number
number
of of
colors
of colors
colors
forfor
for
thethe
the
queen5_5
queen5_5
queen5_5
graph,
graph,
graph,
which
which
which
results
results
results
in in
the
in the
the
global
global
global
solution
solution
solution
of of
five
of five
five
colors.
colors.
colors.
To
To
To
findfind
find
thethe
the
global
global
global
solution,
solution,
solution,
every
every
2ms
2ms
2ms
a pulse
aa pulse
pulse
was
applied.
applied.
applied.
TheThe
The
resulting
resulting
resulting
assignment
assignment
assignment
of of
the
of the
the
vertices
vertices
vertices
to to
different
to different
different
colors,
colors,
colors,
depending
depending
depending
on
Figure
13.
Evidence
for
the every
capability
ofwasawas
25-cell
network,
preliminarily
compensated
for
theon on
their
their
their
final
final
final
phase
phase
phase
shift,
shift,
shift,
is visualized
is
is visualized
visualized
by by
by
highlighting
highlighting
highlighting
of of
the
of the
the
adjacent
adjacent
adjacent
segment
segment
segment
with
with
with
thethe
the
respective
respective
respective
background
background
background
color
color
color
in in
the
in the
the
graph
graph
graph
in in
(b).
in (b).
(b).
Number of colors
Number of colors
G(ϕ)
G(ϕ)
unbalance in the number of connections per oscillator, and for the inter-device variability inherent to
memristors,
to converge toward the optimal solution of the vertex coloring problem for the graph
90°90°
90°
queen5_5
[50]. The45°
cyclic
application
of a pulse stimulus of fixed length
18 18
18 and appropriate amplitude to
135°
135°
135°
45°
45°
60 60
60
an ad-hoc oscillator of the network
16 16
16 the global minimum solution.
40 40
40 guides the phase dynamics toward
100ms
100ms
100ms
80ms
80ms
80ms
60ms
60ms
60ms
20
20
20 time evolution of the phase of each
(a) Phase diagram
visualising
the
oscillator
j ∈ {1, . . . , 24} relative
40ms
40ms
40ms
14 14
14
20ms
20ms
20ms
0ms
0ms
0ms
180°
180°
180°
0° 0°
0°
0 00
to the phase of the reference oscillator
0. (b) Time waveform of the12optimisation
goal function G (ϕ).
12
12
−20−20
−20
(c) Progression of the outcome
of the iterative vertex coloring procedure
of Section 4 over time.
10 10
10
−40−40
−40
225°
225°
225°
315°
315°
315° half of
Throughout
the second
the
−60−60
−60 simulation the minimum number
8 of
88 color groups, assigned to the
0 00 20to
20
20the
40 40
40
60 60
60 80 80
80number
100100
100
00 20 20
20 40 in
40
40 the
60 60
60inset
80 80
80of
100
100
100 (b), is fixed
270°
270°graph queen5_5, 0visualised
vertices of270°
the
plot
chromatic
time
time
time
/ ms
// ms
ms
time
time
time
/ ms
// ms
ms
(a)(a)
(a)
(b)subject to further pulse-based perturbations.
(c)(c)
(c)
n = 5 of the
graph itself, despite the network(b)is(b)
Fig.Fig.
Fig.
17:17:
17:
Evolution
Evolution
Evolution
of of
(a)
of (a)
(a)
thethe
the
phase
phase
phase
shift.
shift.
shift.
(b)(b)
(b)
thethe
the
optimization
optimization
optimization
goal
goal
goal
G(ϕ)
G(ϕ)
G(ϕ)
andand
and
(c)(c)
(c)
thethe
the
number
number
number
of of
colors
of colors
colors
forfor
for
thethe
the
queen6_6
queen6_6
queen6_6
graph,
graph,
graph,
which
which
which
results
results
results
in ain
intemporary
aa temporary
temporary
local
local
local
solution
solution
solution
of of
eight
of eight
eight
colors.
colors.
colors.
To To
To
optimize
optimize
optimize
thethe
the
determined
determined
determined
solution,
solution,
solution,
every
every
every
2ms
2ms
2ms
a crossover
aa crossover
crossover
waswas
was
implemented.
implemented.
implemented.
Table 3 shows a comparison between the solutions of the node coloring task for a
number of graphs [50] pertaining to the 2nd algorithm implementation challenge for NPhardnetwork
problems
Discrete
Mathematics
and Theoretical
Computer
Science
(DIMACS)
[51],
oscillatory
oscillatory
oscillatory
network
network
as as
as
a computing
aaincomputing
computing
engine
engine
engine
forfor
for
thethe
the
implemenimplemenimplemenIEEE
IEEE
IEEE
Transactions
Transactions
Transactions
on on
on
Circuits
Circuits
Circuits
and
and
and
Systems
Systems
Systems
I:
Regular
I:
I: Regular
Regular
Papers,
Papers,
Papers,
vol.vol.
vol.
66,66,
66,
no.no.
no.
7, 2019.
7,
7, 2019.
2019.
tation
tation
tation
of of
of
optimization
optimization
optimization
algorithms.
algorithms.
The
The
The
fact
fact
fact
that,
that,
that,
for
for
for
general
general
general
derived
fromalgorithms.
the
application
of
various
techniques,
namely
an
algorithmic
approach
[2][2]
[2]
R. R.
R.
Tetzlaff,
Tetzlaff,
Tetzlaff,
A. A.
A.
Ascoli,
Ascoli,
Ascoli,
I. Messaris,
I.I. Messaris,
Messaris,
andand
and
L. L.
O.
L. O.
O.
Chua,
Chua,
Chua,
“Theoretical
“Theoretical
“Theoretical
founfounfouninappropriate
inappropriate
inappropriate
initial
initial
initial
conditions,
conditions,
conditions,
thethe
the
network
network
network
converges
converges
converges
to to
to
a aa
dations
dations
dations
of of
memristor
of memristor
memristor
cellular
cellular
cellular
nonlinear
nonlinear
nonlinear
networks:
networks:
networks:
Memcomputing
Memcomputing
Memcomputing
with
with
known
as
Brélaz
heuristic
[52],
methods
based
upon
the
analysis
of
the
phase
dynamics
ofwith
local
local
local
solution,
solution,
solution,
was
was
was
deemed
deemed
deemed
to to
to
bebe
be
thethe
the
main
main
main
reason
reason
reason
why
why
why
the
the
the
bistable-like
bistable-like
bistable-like
memristors,”
memristors,”
memristors,”
IEEE
IEEE
IEEE
Transactions
Transactions
Transactions
on on
on
Circuits
Circuits
Circuits
andand
and
Systems
Systems
Systems
I: I:
I:
Regular
Regular
Regular
Papers,
Papers,
Papers,
2019.
2019.
2019. without a control strategy
capacitively-coupled
arrays
of locally-active
oscillators
system
system
system
diddid
did
rarely
rarely
rarely
determine
determine
determine
thethe
the
graph
graph
graph
chromatic
chromatic
chromatic
number.
number.
number.
ByBy
Bymemristor
[3][3]
[3]
A. A.
A.
Ascoli,
Ascoli,
Ascoli,
I. I.Messaris,
I. Messaris,
Messaris,
R. R.
R.
Tetzlaff,
Tetzlaff,
Tetzlaff,
andand
and
L. L.
L.
O. O.
O.
Chua,
Chua,
Chua,
“Theoretical
“Theoretical
“Theoretical
implementing
implementing
implementing
thethe
the
concepts
concepts
concepts
of of
of
crossovers
crossovers
crossovers
(section
(section
(section
IV-C)
IV-C)
IV-C)
and
and
and
for bypassing
local
minima
solutions,
as
respectively
presented
in nonlinear
[42],
and
in Stability
Section
4,
foundations
foundations
foundations
of of
memristor
of memristor
memristor
cellular
cellular
cellular
nonlinear
nonlinear
networks:
networks:
networks:
Stability
Stability
analysis
analysis
analysis
thethe
thepulse
pulse
pulsedestabilization
destabilization
destabilization(section
(section
(section
IV-D),
IV-D),
IV-D),onon
onthethe
thememristive
memristive
memristive
with
with
with
dynamic
dynamic
dynamic
memristors,”
memristors,”
memristors,”
IEEE
IEEE
IEEE
Transactions
Transactions
Transactions
on on
on
Circuits
Circuits
Circuits
andand
and
Systems
Systems
Systems
and,
finally,
paradigms
including
either
a
reconfigurability
of
the
oscillators’
couplings,
I:
Regular
I:
I:
Regular
Regular
Papers,
Papers,
Papers,
2019.
2019.
2019.
network,
network,
network,
local
local
local
solutions
solutions
solutions
may
may
may
bebe
be
bypassed,
bypassed,
bypassed,
and,
and,
and,
in in
in
most
most
most
graph
graph
graph
[4][4]
[4]
A. A.
A.
Ascoli,
Ascoli,
Ascoli,
R. R.
R.
Tetzlaff,
Tetzlaff,
Tetzlaff,
S.-M.
S.-M.
S.-M.
Kang,
Kang,
Kang,
andand
and
L. L.
L.
O.
O.
O.
Chua,
Chua,
Chua,
“Theoretical
“Theoretical
“Theoretical
as discussed
in Section
5.1,
or
aglobal
perturbation
of
the
memristive
array,
as
presented
in
coloring
coloring
coloring
problems
problems
problems
analyzed
analyzed
analyzed
in in
in
this
this
this
paper,
paper,
paper,
the
the
the
global
global
solution
solution
solution
foundations
foundations
foundations
of of
memristor
of memristor
memristor
cellular
cellular
cellular
nonlinear
nonlinear
nonlinear
networks:
networks:
networks:
AA
DRM
A DRM
DRM
-based
2 -based
22-based
is is
is
finally
finally
finally
attained.
attained.
attained.
The
The
utilization
utilization
utilization
of of
of
these
these
these
concepts
concepts
concepts
tosystem
to
to
solve
solve
solve to method
method
method
to to
to
design
design
design
memcomputers
memcomputers
memcomputers
with
with
dynamic
dynamic
dynamic
memristors,”
memristors,”
memristors,”
IEEE
IEEE
IEEE
Section
5.2,The
to
enable
the
dynamical
exit
an
impasse
state,withand
to resume
the
Transactions
Transactions
Transactions
on on
on
Circuits
Circuits
Circuits
andand
and
Systems
Systems
Systems
I: Regular
I:
I: Regular
Regular
Papers,
Papers,
Papers,
2020.
2020.
2020.
vertex
vertex
vertex
coloring
coloring
coloring
problems
problems
problems
leads
leads
leads
to to
to
a far
aa far
far
superior
superior
superior
performance,
performance,
performance,
calculations
of
the
problem
solution,
thereafter.
[5][5]
[5]
A. A.
A.
Crnkić,
Crnkić,
Crnkić,
J. Povh,
J.
J. Povh,
Povh,
V. V.
Jaćimović,
V. Jaćimović,
Jaćimović,
andand
and
Z. Z.
Levnajić,
Z. Levnajić,
Levnajić,
“Collective
“Collective
“Collective
dynamics
dynamics
dynamics
with
with
with
fewer
fewer
fewer
color
color
color
groups
groups
groups
as as
as
compared
compared
compared
to to
to
those
those
those
identified
identified
identified
viavia
via
of phase-repulsive
of phase-repulsive
phase-repulsive
oscillators
oscillators
oscillators
solves
solves
solves
graph
graph
graph
coloring
coloring
coloring
problem,”
problem,”
problem,”
Chaos:
Chaos:
Chaos:
AnAn
An
The
results
obtained
through
the
analysis
ofof
the
phase
dynamics
in
memristive
osother
other
other
means
means
means
in in
in
other
other
other
significant
significant
significant
studies
studies
studies
reported
reported
reported
recently
recently
recently
in in
in
Interdisciplinary
Interdisciplinary
Interdisciplinary
Journal
Journal
Journal
of of
Nonlinear
of Nonlinear
Nonlinear
Science,
Science,
Science,
vol.vol.
vol.
30,30,
30,
no.no.
no.
3, 2020.
3,
3, 2020.
2020.
[6][6]
[6]
M.M.
M.
K. K.
K.
Bashar,
Bashar,
Bashar,
R. R.
R.
Hrdy,
Hrdy,
Hrdy,
A. A.
A.
Mallick,
Mallick,
Mallick,
F. F.
F.iterative
Hassanzadeh,
F.
F. Hassanzadeh,
Hassanzadeh,
and
and
and
N. N.
N.
Shukla,
Shukla,
Shukla,
thethe
the
literature,
literature,
literature,
as as
as
shown
shown
shown
in in
in
Table
Table
Table
II II
II
from
from
from
section
section
section
V.
V.
cillatory
networks—refer
to
the V.
approach
employed
inthe
[42]
and
to
our
strategy
“Solving
“Solving
“Solving
thethe
maximum
maximum
maximum
independent
independent
independent
setset
set
problem
problem
problem
using
using
using
coupled
coupled
coupled
relaxrelaxrelaxAccording
According
According
to to
to
ourour
our
research
research
research
agenda,
agenda,
agenda,
thethe
the
next
next
next
step
step
step
will
will
will
tackle
tackle
tackle
thethe
the
ation
ation
ation
oscillators,”
oscillators,”
oscillators,”
in
in
2019
in
2019
2019
Device
Device
Device
Research
Research
Research
Conference
Conference
Conference
(DRC).
(DRC).
(DRC).
IEEE,
IEEE,
IEEE,
from Section 4, where no technique to bypass local minima solutions is set in place, are
hardware
hardware
hardware
implementation
implementation
implementation
of of
of
anan
an
array
array
array
of of
of
oscillators
oscillators
oscillators
with
with
with
freely
freely
freely
2019.
2019.
2019.
comparable
to the
corresponding
ones of
the Brélaz
algorithm.
As mayforbe
evinced
[7][7]
[7]
S. S.
S.
A.heuristic
A.
A.
Lee,
Lee,
Lee,
“k-phase
“k-phase
“k-phase
oscillator
oscillator
oscillator
synchronization
synchronization
synchronization
for
for
graph
graph
graph
coloring,”
coloring,”
coloring,”
configurable
configurable
configurable
connections
connections
connections
between
between
between
thethe
the
cells.
cells.
cells.
Furthermore,
Furthermore,
Furthermore,
other
other
other
Mathematics
Mathematics
Mathematics
in in
Computer
in Computer
Computer
Science,
Science,
Science,
vol.vol.
vol.
3, no.
3,
3, no.
no.
1, 2010.
1,
1, 2010.
2010.
concepts
concepts
concepts
or or
or
better
better
better
strategies
strategies
strategies
for
for
thethe
the
given
given
given
concepts,
concepts,
concepts,
allowing
allowing
allowing implemented
from
Table
3,
the for
graph
coloring
paradigm
through
capacitively-coupled
[8][8]
[8]
A. A.
A.
Parihar,
Parihar,
Parihar,
N. N.
N.
Shukla,
Shukla,
Shukla,
M.M.
M.
Jerry,
Jerry,
Jerry,
S. S.
S.
Datta,
Datta,
Datta,
andand
and
A. A.
A.
Raychowdhury,
Raychowdhury,
Raychowdhury,
thethe
the
network
network
network
to to
to
identify
identify
identify
thethe
the
chromatic
chromatic
chromatic
number
number
faster
faster
faster
or or
or
with
with
with vertices
“Computing
“Computing
“Computing
with
with
dynamical
dynamical
dynamical
systems
systems
systems
based
based
based
on on
on
insulator-metal-transition
insulator-metal-transition
insulator-metal-transition
memristive
oscillators
innumber
[42]
classifies
the
of with
all
investigated
graphs,
with the
oscillators,”
oscillators,”
oscillators,”
Nanophotonics,
Nanophotonics,
Nanophotonics,
vol.vol.
vol.
6, no.
6,
6, no.
no.
3, 2017.
3,
3, 2017.
2017.
less
less
lesseffort,
effort,
effort,shall
shall
shallbebe
beinvestigated.
investigated.
investigated.Furthermore,
Furthermore,
Furthermore,thethe
thethermal
thermal
thermal
exception
ofshould
the
one
called
[50], into[9][9]
a[9]A.number
color
groups
equal
to or lower
A.
Parihar,
A. Parihar,
Parihar,
N. N.
Shukla,
N.of
Shukla,
Shukla,
M.M.
M.
Jerry,
Jerry,
Jerry,
S. Datta,
S.
S. Datta,
Datta,
and
and
and
A. A.
Raychowdhury,
A. Raychowdhury,
Raychowdhury,
“Vertex
“Vertex
“Vertex
noise
noise
noise
of of
of
memristors
memristors
memristors
should
should
bebe
be
included
included
included
inqueen6_6
in
in
future
future
future
simulations
simulations
simulations
coloring
coloring
coloring
of of
graphs
of graphs
graphs
viavia
via
phase
phase
phase
dynamics
dynamics
dynamics
of of
coupled
of coupled
coupled
oscillatory
oscillatory
oscillatory
networks,”
networks,”
networks,”
than
theif number
ofhave
colors
assigned
toonthe
nodesScientific
of
the
corresponding
graphs
through
the
to to
todetermine
determine
determine
if
ifit ititmight
might
mighthave
have
a aabeneficial
beneficial
beneficial
effect
effect
effect
on
onthethe
the
Scientific
Scientific
reports,
reports,
reports,
vol.vol.
vol.
7, no.
7,
7, no.
no.
1, 2017.
1,
1, 2017.
2017.
[10]
[10]
[10]
V. V.
Saraswat,
V. Saraswat,
Saraswat,
S. S.
Lashkare,
S. Lashkare,
Lashkare,
P. Kumbhare,
P.
P. Kumbhare,
Kumbhare,
andand
and
U. U.
U.
Ganguly,
Ganguly,
Ganguly,
“Fundamental
“Fundamental
“Fundamental
network
network
network
performance,
performance,
performance,
similarly
similarly
similarly
as as
as
in in
in
a asimulated
afrom
simulated
simulated
annealing
annealing
annealing
proposed
iterative
strategy
Section
4. However,
itnetwork
should
be
pointed
out
that,
while
limit
limit
limit
on
on
on
network
network
size
size
size
scaling
scaling
scaling
of
of
of
oscillatory
oscillatory
oscillatory
neural
neural
neural
networks
networks
networks
duedue
due
to to
to
optimization
optimization
optimization
process.
process.
process.
PrMnO
PrMnO
PrMnO
based
based
based
oscillator
oscillator
oscillator
phase
phase
phase
noise,”
noise,”
noise,”
in
in
2019
in
2019
2019
Device
Device
Device
Research
Research
Research
ConConConthe nominal parameter setting in cell and coupling
circuits
is
unaltered
in
the
numerical
3 33
ference
ference
ference
(DRC).
(DRC).
(DRC).IEEE,
IEEE,
IEEE,
2019.
2019.
2019.
EFERENCES
R
REFERENCES
EFERENCES
investigationsRof
the networks of all the graphs
Table
3,
itLi,
isand
unclear
whether
theof same
[11]
[11]
[11]
J.in
Wu,
J.
J. Wu,
Wu,
L. L.
Jiao,
L. Jiao,
Jiao,
R. R.
R.
Li,
Li,
and
and
W.W.
W.
Chen,
Chen,
Chen,
“Clustering
“Clustering
“Clustering
dynamics
dynamics
dynamics
of
nonlinear
of nonlinear
nonlinear
[1][1]
[1]
M.M.
M.
Weiher,
Weiher,
Weiher,
M.M.
M.
Herzig,
Herzig,
Herzig,
R. R.
Tetzlaff,
R. Tetzlaff,
Tetzlaff,
A. A.
A.
Ascoli,
Ascoli,
Ascoli,
T. Mikolajick,
T.
T. Mikolajick,
Mikolajick,
andand
and
S. SleS.
S. SleSleoscillator
oscillator
oscillator
network:
network:
network:
Application
Application
Application
to to
graph
to graph
graph
coloring
coloring
coloring
problem,”
problem,”
problem,”
Physica
Physica
Physica
D: D:
D:
values
were
assigned
to
the
physical
attributes
of
the
components
of
the
array
for
the
sazeck,
sazeck,
sazeck,
“Pattern
“Pattern
“Pattern
formation
formation
formation
with
with
with
locally
locally
locally
active
active
active
s-type
s-type
s-type
NbO
NbO
NbO
memristors,”
memristors,”
memristors,”
Nonlinear
Nonlinear
Nonlinear
Phenomena,
Phenomena,
Phenomena,
vol.
vol.
vol.
240,
240,
240,
no.
no.
no.
24,
24,
24,
2011.
2011.
2011.
x xx
simulations of the corresponding systems in [42]. Furthermore, while the mathematical
characterisation of our NbOx resistance switching memory is rooted on strong physics foundations, and is endowed with device-to-device variability control, the simplistic, model
adopted to characterise the locally active VO2 memristor in [42], has no physics basis,
assuming that the two-terminal element may feature at any given time one of two conductance values, denoting the metallic and insulating state, respectively, depending upon the
voltage falling across it, and does not account for the inherent spread in the device static
and dynamic properties from sample to sample. The application of our iterative graph
coloring strategy to the vertex orderings derived from the simulations of the networks from
Table 3, for the case where the tendency of the oscillators’ phase shifts to approach local
J. Low Power Electron. Appl. 2022, 12, 22
26 of 30
minima solutions is counterbalanced through the implementation of either the crossover or
the pulse destabilisation control paradigms, leads to an evident performance improvement.
With reference to each of the graphs—namely mycie15, queen5_5, queen6_6, queen7_7,
and queen8_8—whereby, according to the iterative strategy from Section 4, our memristive
network is unable to identify the chromatic number n on its own, the periodic application
of either of the two control paradigms from Sections 5.1 and 5.2 allows the lowest number
of colors assigned to the N vertices to decrease, and, in most cases, the final phase pattern
of the memristive oscillatory array allows to determine the global minimum solution. As an
example, which also reveals how further studies are necessary to improve the performance
of our memristive networks in coloring the vertices of complex graphs, Figure 14(a) shows
the phase dynamics of a balanced network implementing the queen6_6 graph [50], for
the case where the mismatch in the number of couplings per oscillator and the memristor
device-to-device variability are respectively compensated via the methodologies described
in Sections 3.2 and 3.3, and a periodic application of the crossover control paradigm of
Section 5.1, involving a distinct pair of oscillators from cycle to cycle, is set in place. In
this case, the network keeps in a transient phase throughout the simulation. As may be
evinced by inspecting the time evolution of the optimisation goal function G (ϕ), shown
in plot (b), and the evolution of the outcome of our iterative vertex coloring strategy over
time, illustrated in plot (c), the dynamical system does not exhibit a monotonic decrease
toward the global minimum solution, escaping the best solution, computed around 50 ms,
to approach higher local minima thereafter. With regard to our intention to enhance the
local minima bypass paradigms further, the analysis of the potentially-beneficial impact of
the memristor thermal noise source on the capability of the dynamical system to descend
monotonically toward the global minimum solution is one of the future research activities
in our agenda.
Remark 4. The first priority in our research agenda is to realize a hardware prototype able to solve
various graph coloring problems of small/medium size on the basis of the oscillators’ phase dynamics.
In order to allow a hardware implementation of the proposed memristive computing engine to solve
a number of different graph coloring problems, the connections between the processing units need to
be adjustable on a case by case basis, which calls for the use of a coupling arrangement typical of
a Hopfield neural network, with the introduction of a transistor switch, controllable via some ad
hoc multiplexer, in series with each capacitor CC . While hardware architecture considerations for
large networks are still quite premature, we believe that scaling up the size of the computing engine
would require the use of an array-like structure, as typically used for memories, to implement the
programmable coupling circuitry. Except for the memristors, which could be arranged in crossbar
configuration across the metal layers, the rest of the circuitry, necessary to implement the oscillatory
cells, would be laid out on the CMOS substrate. In later generations of the proposed hardware, also
back-end of line (BEOL) transistors can be envisioned so as to further increase the area efficiency
of the computing platform. From a problem-solving perspective, scaling up the network size to
tackle problems of bigger dimension, envisaging, in general, a larger number of connections between
the vertices of the associated graphs, shall result in an inevitable increase in the number of local
minima for the optimization goal function, which complicates the operation of the control circuitry,
as it tries to guide the oscillators’ phases toward the optimal grouping at steady state. This is a
general problem for all state-of-the-art software algorithms and hardware platforms, which aim
to minimize non-convex optimization goal functions. In order to solve graph coloring problems
of higher complexity, some fine tuning of the control strategies, proposed in this manuscript, as
inspired by the most efficient NP-hard optimization problem solvers, available today, is expected to
be necessary.
J. Low Power Electron. Appl. 2022, 12, 22
27 of 30
0
00
Number of colors
Number of colors
0° 0°0°
Number of colors
180°180°
180°
G(ϕ)
G(ϕ)
G(ϕ)
Table 3. Comparison between the solutions of the vertex coloring problem for various graphs [50] for
the 2nd algorithm implementation challenge for NP-hard problems in DIMACS [51], obtained through
the application of a specific algorithm, known as Brélaz heuristic [52], by means of methods exploiting
the
phase
dynamics
of
capacitively-coupled
memristive networks without a control strategy for
IEEEIEEE
TRANSACTIONS
IEEE
TRANSACTIONS
TRANSACTIONS
ON CIRCUITS
ONON
CIRCUITS
CIRCUITS
ANDAND
SYSTEMS—I:
AND
SYSTEMS—I:
SYSTEMS—I:
REGULAR
REGULAR
REGULAR
PAPERS
PAPERS
PAPERS
12 1212
bypassing local minima solutions, namely the technique proposed in [42], and the iterative node
coloring 90°
procedure,
presented in Section 4, and via the iterative node coloring procedure augmented
90°
90°
with
strategies, 45°
based
and pulse destabilisation,14presented
in Sections 5.1 and 5.2,
1414
40crossover
4040
135°135°
135°
45°
45° upon
respectively, and aimed
to
overcome
local
minima
solutions
[31].
The
results
tabulated in the last
12
12
12
100ms
100ms
100ms 20 2020
80ms
80ms
80ms
60ms
60ms
60ms
40ms
40ms
40ms
three columns
were
computed
through
the
analysis
of
100
ms
long
numerical
simulations.
20ms
20ms
20ms
10 1010
0ms
0ms
0ms
8 88
Minimum Number of Color−20
Groups
−20
−20 for the Classification of the Vertices of the Associated Group
6 strategy
66
Brélaz algorithm
[42] iterative strategy iterative strategy and crossover control iterative
and pulse destabilisation control
−40 −40
−40
225°225°
225°
315°
315°
4
4
4 315°
4
4
0
0
0
20
20
20
40
40
40
60
60
60
80
80
80
100
100
100
0
0
0
20
202040 5404060 606080 8080
100 100
100
270° 5
5
5 270°270°
5
time
time
/
time
ms
/
ms
/
ms
time
/ time
ms/ ms
/ ms
6
6
7
6
6 time
7
6 (a) (a)(a)
7
5 (b) (b)(b)
5(c) (c)(c)
Fig.
16:
Fig.
16:
Evolution
16:
Evolution
Evolution
of (a)
of of
the
(a)(a)
phase
thethe
phase
phase
shift.shift.
(b)
shift.
the
(b)(b)
optimization
thethe
optimization
optimization
goalgoal
G(ϕ)
goal
G(ϕ)
and and
(c)and
the
(c)(c)
number
thethe
number
number
of colors
of of
colors
colors
for the
forfor
queen5_5
thethe
queen5_5
queen5_5
graph,
graph,
which
which
which
results
results
results
in the
in in
global
thethe
global
global
10Fig.
12
11
8 G(ϕ)
8graph,
solution
solution
of five
of of
five
colors.
five
colors.
To find
ToTo
find
the
find
global
thethe
global
global
solution,
solution,
solution,
every
every
every
2ms2ms
a 2ms
pulse
a pulse
a was
pulse
applied.
was
applied.
applied.
The The
resulting
The
resulting
resulting
assignment
assignment
assignment
of the
of of
vertices
thethe
vertices
vertices
to 10
different
to to
different
different
colors,
colors,
colors,
depending
depending
depending
on onon
12solution
12colors.
14
10 was
their
final
their
final
phase
final
phase
phase
shift,
is
shift,
visualized
is is
visualized
visualized
byby
highlighting
highlighting
of the
of of
adjacent
thethe
adjacent
adjacent
segment
segment
segment
withwith
the
with
respective
thethe
respective
respective
background
background
background
colorcolor
in
color
the
in in
graph
thethe
graph
graph
in (b).
in in
(b).
(b).
15their
14shift,
15by highlighting
13
13
90° 90°
90°
180°180°
180°
45° 45°
45°
100ms
100ms
100ms
80ms
80ms
80ms
60ms
60ms
60ms
40ms
40ms
40ms
20ms
20ms
20ms
0ms
0ms
0ms
0° 0°0°
60 6060
18 1818
40 4040
16 1616
20 2020
0
00
−20 −20
−20
−40 −40
−40
225°225°
225°
315°315°
315°
270°270°
270°
(a) (a)(a)
−60 −60
−60
0 0 020 202040 404060 606080 8080
100 100
100
timetime
/ time
ms/ ms
/ ms
(b) (b)(b)
Number of colors
Number of colors
135°135°
135°
Number of colors
n
4
5
6
5
7
7
9
G(ϕ)
G(ϕ)
vertices
11
20
47
25
36
49
64
G(ϕ)
graph
mycie13
mycie14
mycie15
queen5_5
queen6_6
queen7_7
queen8_8
14 1414
12 1212
10 1010
8
88
0 0 020 202040 404060 606080 8080
100 100
100
timetime
/ time
ms/ ms
/ ms
(c) (c)(c)
Fig.Fig.
17:
Fig.
Evolution
17:17:
Evolution
Evolution
of (a)
of of
the
(a)(a)
phase
thethe
phase
phase
shift.shift.
(b)
shift.
the
(b)(b)
optimization
thethe
optimization
optimization
goalgoal
G(ϕ)
goal
G(ϕ)
G(ϕ)
and and
(c)and
the
(c)(c)
number
thethe
number
number
of colors
of of
colors
colors
for the
forfor
queen6_6
thethe
queen6_6
queen6_6
graph,
graph,
graph,
which
which
which
results
results
results
in a in
temporary
in
a temporary
a temporary
locallocal
solution
local
solution
solution
of eight
of of
eight
colors.
eight
colors.
colors.
To Phase
optimize
ToTo
optimize
optimize
the
determined
thethe
determined
determined
solution,
solution,
every
every
every
2ms2ms
a 2ms
crossover
a associated
crossover
a crossover
was was
implemented.
was
implemented.
implemented.
Figure
14.
(a)
dynamics
ofsolution,
the
network
to the graph queen6_6 [50], after its
preliminary compensation for the unbalance in the number of connections per oscillator, and for the
inter-device
inherent
memristors,
upon
the
periodic
between
the
couplings
oscillatory
oscillatory
oscillatory
network
network
network
as variability
aascomputing
asa acomputing
computing
engine
engine
engine
forto
for
the
for
the
implementheimplemenimplemenIEEE
IEEE
IEEE
Transactions
Transactions
Transactions
oninterchange
Circuits
onon
Circuits
Circuits
and and
Systems
and
Systems
Systems
I: Regular
I: I:
Regular
Regular
Papers,
Papers,
Papers,
vol. vol.
66,
vol.
66,66,
no. 7,
no.no.
2019.
7, 7,
2019.
2019.
tation
tation
tation
of
optimization
ofoptimization
optimization
algorithms.
algorithms.
algorithms.
TheThe
The
factthis
fact
that,
factthat,
that,
for for
general
forgeneral
general dynamics
ofof
two
specific
oscillators.
In
case
the
phase
of the network keep in a transient state
[2] [2]
R.[2]
Tetzlaff,
R.R.
Tetzlaff,
Tetzlaff,
A. Ascoli,
A.A.
Ascoli,
Ascoli,
I. Messaris,
I. I.
Messaris,
Messaris,
and and
L.and
O.
L. L.
Chua,
O.O.
Chua,
Chua,
“Theoretical
“Theoretical
“Theoretical
founfounfouninappropriate
inappropriate
inappropriate
initial
initial
initial
conditions,
conditions,
conditions,
the the
network
thenetwork
network
converges
converges
converges
to atoEvolution
toa a dations
dations
dations
of the
memristor
of of
memristor
memristor
cellular
cellular
cellular
nonlinear
nonlinear
nonlinear
networks:
networks:
networks:
Memcomputing
Memcomputing
Memcomputing
withwith
with
throughout
the
100 ms-long
simulation.
(b)
of
optimisation
goal
function
G (ϕ) over
local
local
local
solution,
solution,
solution,
waswas
was
deemed
deemed
deemed
to be
totobe
the
bethe
main
themain
main
reason
reason
reason
whywhy
why
the the
the bistable-like
bistable-like
bistable-like
memristors,”
memristors,”
memristors,”
IEEE
IEEE
Transactions
IEEE
Transactions
Transactions
on Circuits
onon
Circuits
Circuits
and and
Systems
and
Systems
Systems
I: I: I:
time.
(c)
Minimum
number
of
colors,
assigned
to
the
nodes
of
the
graph
queen6_6
through
the
Regular
Regular
Regular
Papers,
Papers,
Papers,
2019.
2019.
2019.
system
system
system
diddid
rarely
didrarely
rarely
determine
determine
determine
the the
graph
thegraph
graph
chromatic
chromatic
chromatic
number.
number.
number.
By ByBy
[3] [3]
A.[3]Ascoli,
A.A.Ascoli,
Ascoli,
I. Messaris,
I. I.Messaris,
Messaris,
R. Tetzlaff,
R.R.Tetzlaff,
Tetzlaff,
and and
L.andO.
L. L.Chua,
O.O.Chua,
Chua,
“Theoretical
“Theoretical
“Theoretical
implementing
implementing
implementing
the the
concepts
theconcepts
concepts
of crossovers
ofofcrossovers
crossovers
(section
(section
(section
IV-C)
IV-C)
andand
and
iterative
vertex
coloring
procedure
of IV-C)
Section
4, applied
once
every
cycle,
versus
time.
Half
way
foundations
foundations
foundations
of memristor
of of
memristor
memristor
cellular
cellular
cellular
nonlinear
nonlinear
nonlinear
networks:
networks:
networks:
Stability
Stability
Stability
analysis
analysis
analysis
the the
pulse
thethrough
pulse
pulse
destabilization
destabilization
destabilization
(section
(section
(section
IV-D),
IV-D),
IV-D),
on
on
the
on
the
memristive
the
memristive
memristive
withwith
dynamic
with
dynamic
dynamic
memristors,”
memristors,”
memristors,”
IEEE
IEEE
IEEE
Transactions
Transactions
Transactions
on Circuits
onon
Circuits
Circuits
and and
Systems
and
Systems
Systems
the simulation the nodes of the graph, illustrated
in
the
inset
of
plot
(b),
are
classified
into
I: I:
Regular
Regular
Papers,
Papers,
Papers,
2019.
2019.
2019.
network,
network,
network,
local
local
local
solutions
solutions
solutions
maymay
may
be bypassed,
bebebypassed,
bypassed,
and,and,
and,
in most
ininmost
most
graph
graph
graph I: Regular
[4]
[4]
A.
[4]
Ascoli,
A.
A.
Ascoli,
Ascoli,
R.
Tetzlaff,
R.
R.
Tetzlaff,
Tetzlaff,
S.-M.
S.-M.
S.-M.
Kang,
Kang,
Kang,
and
and
L.
and
O.
L.
L.
Chua,
O.
O.
Chua,
Chua,
“Theoretical
“Theoretical
“Theoretical
8 problems
color
groups,
one
more
than
correct
number (refer to Table 3), but this solution proves to be
coloring
coloring
coloring
problems
problems
analyzed
analyzed
analyzed
in this
in
inthis
paper,
thispaper,
paper,
thethe
the
global
theglobal
global
solution
solution
solution
foundations
foundations
foundations
of memristor
of of
memristor
memristor
cellular
cellular
cellular
nonlinear
nonlinear
nonlinear
networks:
networks:
networks:
A DRM
AA
DRM
-based
2DRM
2 -based
2 -based
is finally
is isfinally
finally
attained.
attained.
attained.
TheThe
utilization
The
utilization
utilization
of these
ofofthese
these
concepts
concepts
concepts
to issolve
tosubject
tosolve
solve tomethod
unstable,
when
the
network,
thereafter,
further
perturbations.
method
method
to design
tocrossover-based
todesign
design
memcomputers
memcomputers
memcomputers
with
with
dynamic
withdynamic
dynamic
memristors,”
memristors,”
memristors,”
IEEE
IEEE
IEEE
Transactions
Transactions
on Circuits
onon
Circuits
Circuits
and and
Systems
and
Systems
Systems
I: Regular
I: I:
Regular
Regular
Papers,
Papers,
Papers,
2020.
2020.
2020.
vertex
vertex
vertex
coloring
coloring
coloring
problems
problems
problems
leads
leads
leads
to atofar
toa afar
superior
farsuperior
superior
performance,
performance,
performance, Transactions
A.[5]
Crnkić,
A.A.
Crnkić,
Crnkić,
J. Povh,
J. J.
Povh,
Povh,
V. Jaćimović,
V. V.
Jaćimović,
Jaćimović,
and and
Z.and
Levnajić,
Z. Z.
Levnajić,
Levnajić,
“Collective
“Collective
“Collective
dynamics
dynamics
dynamics
withwith
with
fewer
fewer
color
color
color
groups
groups
groups
as compared
asascompared
compared
to those
totothose
those
identified
identified
identified
via via
via[5] [5]
6.fewer
Conclusions
of phase-repulsive
of of
phase-repulsive
phase-repulsive
oscillators
oscillators
oscillators
solves
solves
solves
graph
graph
graph
coloring
coloring
coloring
problem,”
problem,”
problem,”
Chaos:
Chaos:
Chaos:
An AnAn
other
other
other
means
means
means
in other
ininother
other
significant
significant
significant
studies
studies
studies
reported
reported
reported
recently
recently
recently
in inin Interdisciplinary
Interdisciplinary
Interdisciplinary
Journal
Journal
Journal
of Nonlinear
of of
Nonlinear
Nonlinear
Science,
Science,
Science,
vol. vol.
30,
vol.
no.
30,30,
3,
no.no.
2020.
3, 3,
2020.
2020.
The
local
of
NbO
emulation
ofandneuronal
[6] [6]
M.[6]([27,28])
K.
M.M.
Bashar,
K.K.
Bashar,
Bashar,
R. allows
Hrdy,
R.R.
Hrdy,
Hrdy,
A. Mallick,
A.the
A.
Mallick,
Mallick,
F. F.F.Hassanzadeh,
F.
F. F.
Hassanzadeh,
Hassanzadeh,
and
N.and
Shukla,
N.N.
Shukla,
Shukla,
the the
literature,
theliterature,
literature,
as shown
asas
shown
shown
in activity
Table
ininTable
Table
II from
II[26]
IIfrom
from
section
section
section
V. V.xV.memristors
“Solving
“Solving
“Solving
the the
maximum
themaximum
maximum
independent
independent
independent
set problem
setsetproblem
problem
using
using
using
coupled
coupled
coupled
relaxrelaxrelaxAccording
According
According
to our
totoour
research
ourresearch
research
agenda,
agenda,
agenda,
the the
next
thenext
step
nextstep
will
stepwill
tackle
willtackle
tackle
theof
the
the
dynamics
([9,11]),
the
implementation
bio-inspired
signal
processing
paradigms
[53],
ationation
oscillators,”
ation
oscillators,”
oscillators,”
in 2019
in in
2019
Device
2019
Device
Device
Research
Research
Research
Conference
Conference
Conference
(DRC).
(DRC).
(DRC).
IEEE,
IEEE,
IEEE,
hardware
hardware
hardware
implementation
implementation
implementation
of an
ofofarray
ananarray
array
of oscillators
ofofoscillators
oscillators
withwith
with
freely
freely
freely 2019.
2019.
2019. emerging in systems from cellular
and the reproduction of complex phenomena
[30]
S.[7]A.
S. S.Lee,
A.A.Lee,
“k-phase
Lee,“k-phase
“k-phase
oscillator
oscillator
oscillator
synchronization
synchronization
synchronization
for for
graph
forgraph
graph
coloring,”
coloring,”
coloring,”
configurable
configurable
configurable
connections
connections
connections
between
between
between
the the
cells.
thecells.
cells.
Furthermore,
Furthermore,
Furthermore,
other
other
other[7] [7]
Mathematics
Mathematics
Mathematics
intutorial
Computer
in in
Computer
Computer
Science,
Science,
vol.operating
vol.
3,vol.
no.
3, 3,
1,
no.no.
2010.
1, 1,
2010.
2010.
biology
[44].
This
manuscript
serves
as
a
pedagogical
toScience,
the
principles
concepts
concepts
concepts
or better
ororbetter
better
strategies
strategies
strategies
for for
the
forthe
given
thegiven
given
concepts,
concepts,
concepts,
allowing
allowing
allowing[8] [8]
A.[8]Parihar,
A.A.Parihar,
Parihar,
N. Shukla,
N.N.Shukla,
Shukla,
M. M.
Jerry,
M.Jerry,
Jerry,
S. Datta,
S. S.Datta,
Datta,
and and
A.
andRaychowdhury,
A.A.Raychowdhury,
Raychowdhury,
the the
network
thenetwork
identify
totoidentify
identify
the
the
chromatic
thechromatic
chromatic
number
number
number
faster
faster
faster
with
ororwith
with “Computing
ofnetwork
atocellular
nonlinear
network
ofor oscillators,
coupled
through
linear
capacitors,
and
“Computing
“Computing
withwith
dynamical
with
dynamical
dynamical
systems
systems
systems
based
based
based
on insulator-metal-transition
onon
insulator-metal-transition
insulator-metal-transition
oscillators,”
oscillators,”
oscillators,”
Nanophotonics,
Nanophotonics,
Nanophotonics,
vol. vol.
6,vol.
no.
6, 6,
3,
no.no.
2017.
3, 3,
2017.
2017.
lessless
less
effort,
effort,
effort,
shall
shall
shall
be be
investigated.
be
investigated.
investigated.
Furthermore,
Furthermore,
Furthermore,
the the
thermal
thethermal
thermal
employing
one
locally-active
memristor
[43]
each,
recently
introduced
in
[31]
to “Vertex
solve
[9]
[9]
A.
[9]
Parihar,
A.
A.
Parihar,
Parihar,
N.
Shukla,
N.
N.
Shukla,
Shukla,
M.
Jerry,
M.
M.
Jerry,
S.
Jerry,
Datta,
S.
S.
Datta,
Datta,
and
and
A.
and
Raychowdhury,
A.
A.
Raychowdhury,
Raychowdhury,
“Vertex
“Vertex
noise
noise
noise
of memristors
ofofmemristors
memristors
should
should
should
be be
included
beincluded
included
in future
ininfuture
future
simulations
simulations
simulations
coloring
coloring
coloring
of graphs
of of
graphs
graphs
via optimization
phase
viavia
phase
phase
dynamics
dynamics
dynamics
of coupled
of of
coupled
coupled
oscillatory
oscillatory
oscillatory
networks,”
networks,”
networks,”
a
non-deterministic
polynomial
(NP)-hard
combinatorial
problem,
known
to to
determine
todetermine
determine
if it
if ifmight
it itmight
might
have
have
have
a beneficial
a abeneficial
beneficial
effect
effect
effect
on on
the
onthe
the Scientific
Scientific
Scientific
reports,
reports,
reports,
vol. vol.
7,vol.
no.
7, 7,
1,
no.no.
2017.
1, 1,
2017.
2017.
[10][10]
V.
[10]
Saraswat,
V. V.
Saraswat,
Saraswat,
S.
Lashkare,
S. S.
Lashkare,
Lashkare,
Kumbhare,
P. P.
Kumbhare,
Kumbhare,
and
and
U.and
Ganguly,
U.U.
Ganguly,
Ganguly,
“Fundamental
“Fundamental
“Fundamental
asperformance,
vertex
coloring.
due
toannealing
page
limitation,
only
a P.compact
description
of
the
network
network
network
performance,
performance,
similarly
similarly
similarly
as While,
as
inasainin
simulated
a asimulated
simulated
annealing
annealing
limitlimit
on
limitnetwork
ononnetwork
network
size size
scaling
sizescaling
scaling
of oscillatory
of ofoscillatory
oscillatory
neural
neural
neural
networks
networks
networks
due due
todueto to
optimization
optimization
optimization
process.
process.
process.
signal
processing
paradigm, implemented by the proposed Memristor Oscillatory Network,
PrMnO
PrMnO
PrMnO
oscillator
oscillator
oscillator
phase
phase
phase
noise,”
noise,”
noise,”
in 2019
in in2019
Device
2019Device
Device
Research
Research
Research
Con-ConCon3 based
3 based
3 based
ference
ference
ference
(DRC).
(DRC).
(DRC).
IEEE,
IEEE,
IEEE,
2019.
2019.
2019.
R EFERENCES
R EFERENCES
R[31],
EFERENCES
was reported
in
this tutorial reports[11]
all[11]
the
details
of
the
mechanisms
underlying
J.[11]
Wu,
J. J.
Wu,
L.Wu,
Jiao,
L. L.
Jiao,
R.
Jiao,
Li,
R.R.
and
Li,Li,
and
W.and
Chen,
W.W.
Chen,
Chen,
“Clustering
“Clustering
“Clustering
dynamics
dynamics
dynamics
of nonlinear
of of
nonlinear
nonlinear
[1] [1]
M.[1]Weiher,
M.M.
Weiher,
Weiher,
M. Herzig,
M.M.
Herzig,
Herzig,
R. Tetzlaff,
R.R.
Tetzlaff,
Tetzlaff,
A. Ascoli,
A.A.
Ascoli,
Ascoli,
T. Mikolajick,
T. T.
Mikolajick,
Mikolajick,
and
and
S.and
SleS. S.
SleSleoscillator
oscillator
oscillator
network:
network:
network:
Application
Application
Application
to graph
to to
graph
graph
coloring
coloring
coloring
problem,”
problem,”
problem,”
Physica
Physica
Physica
D: D:D:
its
modus
operandi.
Importantly,
control
methods
[33]
to
compensate
for
the
inherent
sazeck,
sazeck,
sazeck,
“Pattern
“Pattern
“Pattern
formation
formation
formation
withwith
locally
with
locally
locally
active
active
active
s-type
s-type
s-type
NbONbO
memristors,”
Nonlinear
Nonlinear
Nonlinear
Phenomena,
Phenomena,
Phenomena,
vol. vol.
240,
vol.
240,
no.
240,
24,
no.no.
2011.
24,24,
2011.
2011.
x NbO
x memristors,”
x memristors,”
variability of memristor devices, to counteract the imbalance between the load capacitances
of the oscillators, as well as, most importantly, to prevent the bio-inspired network to
attain a sub-optimal steady state, are developed and implemented in circuit form. The
Memristor Oscillatory Network, endowed with the proposed control circuitry, is found to
outperform state-of-the-art software and hardware competitor alternatives, identifying, for
J. Low Power Electron. Appl. 2022, 12, 22
28 of 30
each graph from a wide selection, the lowest number of color groups for the respective
vertices. As a more general conclusion, the potential of all locally-active devices, including
niobium ([28,43,54]) or vanadium dioxide [55] threshold switches, and ovonic threshold
switches [56,57], is expected to be subject to a thorough exploration, in the years to come,
for a possible exploitation of their small-signal amplification capability for electronics
applications, e.g. to build nano-oscillators with tuneable frequency ([58]), to solve NP-hard
combinatorial optimization problems, as discussed in this manuscript, for reproducing
complex biological phenomena [44], for exploring new forms of computing via pattern
formation dynamics [59], or for designing bio-plausible neuromorphic circuits [10].
Author Contributions: All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.
Data Availability Statement: Not applicable.
Conflicts of Interest: The authors declare no conflict of interest.
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
Chua, L.O. Memristor: The missing circuit element. IEEE Trans. Circuit Theory 1971, 18, 507–519. [CrossRef]
Chua, L.O.; Kang, S. Memristive devices and systems. Proc. IEEE 1976, 64, 209–223. [CrossRef]
Strukov, D.B.; Snider, G.S.; Stewart, D.R.; Williams, R.S. The missing memristor found. Nature 2008, 453, 80–83. [CrossRef]
[PubMed]
Prodomakis, T.; Toumazou, C.; Chua, L.O. Two Centuries of Memristors. Nat. Mater. 2012, 11, 478–481. [CrossRef] [PubMed]
Chua, L.O. If It’s Pinched, It’s a Memristor. Semicond. Sci. Technol. 2014, 29, 104001. [CrossRef]
Ielmini, D.; Waser, R. Resistive Switching: From Fundamentals of Nanoionic Redox Processes to Memristive Device Applications, 1st ed.;
Wiley-VCH: Weinheim, Germany, 2016; ISBN-13: 978-3527334179.
Mikolajick, T.; Salinga, M.; Kund, M.; Kever, T. Nonvolatile Memory Concepts Based on Resistive Switching in Inorganic Materials.
Adv. Eng. Mater. 2009, 11, 235–240. [CrossRef]
Indiveri, G.; Linares-Barranco, B.; Legenstein, R.; Deligeorgis, G.; Prodromakis, T. Integration of nanoscale memristor synapses in
neuromorphic computing architectures. Nanotechnology 2013, 24, 384010. [CrossRef]
Pickett, M.D.; Medeiros-Ribeiro, G.; Williams, R.S. A scalable neuristor built with Mott memristors. Nat. Mater. 2013, 12, 114–117.
[CrossRef]
Yi, W.; Tsang, K.K.; Lam, S.K.; Bai, X.; Crowell, J.A.; Flores, E.A. Biological plausibility and stochasticity in scalable VO2 active
memristor neurons. Nat. Commun. 2018, 9, 1–10. [CrossRef]
Kang, S.M.; Choi, D.; Eshraghian, J.K.; Zhou, P.; Kim, J.; Kong, B.S.; Zhu, X.; Demirkol, A.S.; Ascoli, A.; Tetzlaff, R.; Lu, W.D.;
Chua, L.O. How to Build a Memristive Integrate-and-Fire Model for Spiking Neuronal Signal Generation. IEEE Trans. Circuits
Syst. I Regul. Pap. 2021, 68, 4837–4850. [CrossRef]
Tetzlaff, R.; Ascoli, A.; Messaris, I.; Chua, L.O. Theoretical Foundations of Memristor Cellular Nonlinear Networks: Memcomputing
with Bistable-like Memristors. IEEE Trans. Circuits Syst. I Regul. Pap. 2020, 67, 502–515. [CrossRef]
Ascoli, A.; Messaris, I.; Tetzlaff, R.; Chua, L.O. Theoretical Foundations of Memristor Cellular Nonlinear Networks: Stability Analysis
with Dynamic Memristors. IEEE Trans. Circuits Syst. I Regul. Pap. 2020, 67, 1389–1401. [CrossRef]
Ascoli, A.; Tetzlaff, R.; Kang, S.M.; Chua, L.O. Theoretical Foundations of Memristor Cellular Nonlinear Networks: A DRM2 based Method to Design Memcomputers with Dynamic Memristors. IEEE Trans. Circuits Syst. I Regul. Pap. 2020, 67, 2753–2766.
[CrossRef]
Chua, L.O. (Ed.) CNN: A Paradigm for Complexity; World Scientific Series on Nonlinear Science: Singapore, 1998; ISBN 9789810234836.
Xia, Q.; Yang, J.J. Memristive crossbar arrays for brain-inspired computing. Nat. Mater. 2019, 18, 309–323. [CrossRef] [PubMed]
Ventra, M.D.; Traversa, F.L. Perspective: Memcomputing: Leveraging memory and physics to compute efficiently. J. Appl. Phys.
2018, 123, 180901. [CrossRef]
Talati, N.; Gupta, S.; Mane, P.; Kvatinsky, S. Logic Design within Memristive Memories Using Memristor Aided loGIC (MAGIC).
IEEE Trans. Nanotechnol. 2016, 15, 635–650. [CrossRef]
Ali, A.H.; Hur, R.B.; Wald, N.; Ronen, R.; Kvatinsky, S. Not in Name Alone: A Memristive Memory Processing Unit for Real
In-Memory Processing. IEEE Micro 2018, 38, 13–21.
Ielmini, D.; Wong, H.-S.P. In-memory computing with resistive switching devices. Nat. Electron. 2018, 1, 333–343. [CrossRef]
Tzouvadaki, I.; Jolly, P.; Lu, X.; Ingebrandt, S.; de Micheli, G.; Estrela, P.; Carrara, S. Label-Free Ultrasensitive Memristive
Aptasensor. Nanoletters 2016, 16, 4472–4476. [CrossRef]
Ibarlucea, B.; Akbar, T.F.; Kim, K.; Rim, T.; Baek, C.-K.; Ascoli, A.; Tetzlaff, R.; Baraban, L.; Cuniberti, G. Ultrasensitive Detection
of Ebola Matrix Protein in a memristor mode. NanoResearch 2018, 11, 1057–1068. [CrossRef]
J. Low Power Electron. Appl. 2022, 12, 22
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
29 of 30
Wang, Z.; Joshi, S.; Savel’ev, S.; Song, W.; Midya, R.; Li, Y.; Rao, M.; Yan, P.; Asapu, S.; Zhuo, Y.; et al. Fully memristive neural
networks for pattern classification with unsupervised learning. Nat. Electron. 2018, 1, 137–145. [CrossRef]
Sebastian, A.; Tuma, T.; Papandreou, N.; Gallo, M.L.; Kull, L.; Parnell, T.; Eleftheriou, E. Temporal correlation detection using
computational phase-change memory. Nat. Commun. 2017, 8, 1115. [CrossRef] [PubMed]
Sheng, X.; Graves, C.E.; Kumar, S.; Li, X.; Buchanan, B.; Zheng, L.; Lam, S.; Li, C.; Strachan, J.P. Low-Conductance and Multilevel
CMOS-Integrated Nanoscale Oxide Memristors. Adv. Electron. Mater. 2019, 5, 1800876. [CrossRef]
Chua, L.O. Local activity is the origin of complexity. Int. J. Bifurc. Chaos 2005, 15, 3435–3456. [CrossRef]
Pickett, M.D.; Williams, R.S. Sub-100 fJ and sub-nanosecond thermally driven threshold switching in niobium oxide crosspoint
nanodevices. Nanotechnology 2012, 23, 215202. [CrossRef]
Ascoli, A.; Slesazeck, S.; Mähne, H.; Tetzlaff, R.; Mikolajick, T. Nonlinear dynamics of a locally-active memristor. IEEE Trans.
Circuits Syst. I (TCAS–I) Regul. Pap. 2015, 62, 1165–1174. [CrossRef]
Cai, F.; Kumar, S.; Vaerenbergh, T.V.; Sheng, X.; Liu, R.; Li, C.; Liu, Z.; Foltin, M.; Yu, S.; Xia, Q.; Yang, J.J.; Beausoleil, R.; Lu,
W.D.; Strachan, J.P. Power-efficient combinatorial optimization using intrinsic noise in memristor Hopfield neural networks. Nat.
Electron. 2020, 3, 409–418. [CrossRef]
Weiher, M.; Herzig, M.; Tetzlaff, R.; Ascoli, A.; Mikolajick, T.; Slesazeck, S. Pattern formation with local active S-type NbOx
memristors. IEEE Trans. Circuits Syst. I Regul. Pap. 2019, 66, 2627–2638. [CrossRef]
Weiher, M.; Herzig, M.; Tetzlaff, R.; Ascoli, A.; Slesazeck, S.; Mikolajick, T. Improved Vertex Coloring With NbOx Memristor-Based
Oscillatory Networks. IEEE Trans. Circuits Syst. I 2021, 68, 2082–2095. [CrossRef]
Vázquez, A.R.; Fernández-Berni, J.; Leñero-Bardallo, J.A.; Vornicu, I.; Carmona-Galán, R. CMOS Vision Sensors: Embedding
Computer Vision at Imaging Front-Ends. IEEE Circuits Syst. Mag. 2018, 18, 90–107. [CrossRef]
Ascoli, A.; Weiher, M.; Herzig, M.; Tetzlaff, R.; Slesazeck, S.; Mikolajick, T. Control Strategies to Optimize Graph Coloring via
M-CNNs with Locally-Active NbOx Memristors. In Proceedings of the International Conference on Modern Circuits and Systems
Technologies (MOCAST) on Electronics and Communications, Thessaloniki, Greece, 5–7 July 2021.
Slesazeck, S.; Mähne, H.; Wylezich, H.; Wachowiak, A.; Radhakrishnan, J.; Ascoli, A.; Tetzlaff, R.; Mikolajick, T. Physical model of
threshold switching in NbO2 based memristors. J. R. Soc. Chem. 2015, 5, 102318–102322. [CrossRef]
Slesazeck, S.; Herzig, M.; Mikolajick, T.; Ascoli, A.; Weiher, M.; Tetzlaff, R. Analysis of Vth variability in NbOx -based threshold
switches. In Proceedings of the IEEE Nonvolatile Memory Technology Symposium (NVMTS), Pittsburgh, PA, USA, 17–19 October
2016; Carnegie Mellon University: Pittsburgh, PA, USA, 2016. [CrossRef]
Gibson, G.A.; Musunuru, S.; Zhang, J.; Vandenberghe, K.; Lee, J.; Hsieh, C.-C.; Jackson, W.; Jeon, Y.; Henze, D.; Li, Z.; Williams,
R.S. An accurate locally active memristor model for S-type negative differential resistance in NbOx . Appl. Phys. Lett. 2016, 108,
023505. [CrossRef]
Herzig, M.; Weiher, M.; Ascoli, A.; Tetzlaff, R.; Mikolajick, T.; Slesazeck, S. Multiple slopes in the negative differential resistance
region of NbOx -based threshold switches. J. Phys. D Appl. Phys. 2019, 52, 325104. [CrossRef]
Herzig, M.; Weiher, M.; Ascoli, A.; Tetzlaff, R.; Mikolajick, T.; Slesazeck, S. Improvement of NbOx -based threshold switching
devices by implementing multilayer stacks. Semicond. Sci. Technol. 2019, 34, 075005. [CrossRef]
Pickett, M.D.; Strukov, D.B.; Borghetti, J.L.; Yang, J.J.; Snider, G.S.; Stewart, D.R.; Williams, R.S. Switching dynamics in titanium
dioxide memristive devices. J. Appl. Phys. 2009, 106, 074508. [CrossRef]
Chua, L.O. Five Non-Volatile Memristor Enigmas Solved. Appl. Phys. A 2018, 124, 563. [CrossRef]
Wu, C.W. Graph Coloring via Synchronization of Coupled Oscillators. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 1998, 45,
974–978.
Parihar, A.; Shukla, N.; Jerry, M.; Datta, S.; Raychowdhury, A. Vertex coloring of graphs via phase dynamics of coupled oscillatory
networks. Sci. Rep. 2017, 7, 1–11. [CrossRef]
Ascoli, A.; Demirkol, A.S.; Tetzlaff, R.; Slesazeck, S.; Mikolajick, T.; Chua, L.O. On Local Activity and Edge of Chaos in a NaMLab
Memristor. Front. Neurosci. 2021, 15. [CrossRef]
Ascoli, A.; Demirkol, A.S.; Tetzlaff, R.; Chua, L.O. Edge of Chaos Theory Resolves Smale Paradox. IEEE Trans. Circuits Syst. I Reg.
Pap. 2022, 69, 252–1265. [CrossRef]
Mitchell, M. An Introduction to Genetic Algorithms; MIT Press: Cambridge, MA, USA, 1998; ISBN 978-0262631853.
Chibante, R. Simulated Annealing: Theory with Applications; Sciyo: Rijeka, Croatia, 2010; ISBN 978-953-307-134-3.
Ascoli, A.; Baumann, D.; Tetzlaff, R.; Chua, L.O.; Hild, M. Memristor-enhanced humanoid robot control system–Part I: Theory
behind the novel memcomputing paradigm. Int. J. Circuit Theory Appl. IJCTA 2018, 46, 155–183. [CrossRef]
Baumann, D.; Ascoli, A.; Tetzlaff, R.; Chua, L.O.; Hild, M. Memristor-enhanced humanoid robot control system–Part II: Circuit
theoretic model and performance analysis. Int. J. Circuit Theory Appl. IJCTA 2018, 46, 184–220. [CrossRef]
Sharma, A.A.; Bain, J.A.; Weldon, J.A. Phase coupling and control of oxide-based oscillators for neuromorphic computing. IEEE J.
Explor. Solid-State Comput. Devices Circuits 2015, 1, 58–66. [CrossRef]
Graph Coloring Instances. Available online: https://mat.tepper.cmu.edu/COLOR/instances.html (accessed on 25 March 2022).
Johnson, D.S.; Trick, M.A. Cliques, Coloring, and Satisfiability: Second DIMACS Implementation Challenge; Based upon the Proceedings
of the DIMACS Workshop, 11–13 October 1993; Series in Discrete Mathematics and Theoretical Computer Science; American
Mathematical Society: Providence, RI, USA, 1996; Volume 26.
Brélaz, D. New Methods to Color the Vertices of a Graph. Commun. Assoc. Comput. Mach. ACM 1979, 22, 251–256. [CrossRef]
J. Low Power Electron. Appl. 2022, 12, 22
53.
54.
55.
56.
57.
58.
59.
30 of 30
Pickett, M.D.; Williams, R.S. Phase transitions enable computational universality in neuristor-based cellular automata. Nanotechnology 2013, 24, 384002. [CrossRef]
Messaris, I.; Brown, T.D.; Demirkol, A.S.; Ascoli, A.; Chawa, M.M.A.; Williams, R.S.; Tetzlaff, R.; Chua, L.O. NbO2 -Mott Memristor:
A Circuit-Theoretic Investigation. IEEE Trans. Circuits Syst. I Regul. Pap. 2021, 68, 4979–4992. [CrossRef]
Liu, X.; Zhang, P.; Nath, S.K.; Li, S.; Nandi, S.K.; Elliman, R.G. Understanding composite negative differential resistance in
niobium oxide memristors. J. Phys. D Appl. Phys. 2022, 55, 105106. [CrossRef]
Callarotti, R.C.; Schmidt, P.E. Theoretical and experimental study of the operation of ovonic switches in the relaxation oscillation
mode. I. The charging characteristic during the off state. J. Appl. Phys. 1984, 55, 3144. [CrossRef]
Callarotti, R.C.; Schmidt, P.E. Theoretical and experimental study of the operation of ovonic switches in the relaxation oscillation
mode. II. The discharging characteristics and the equivalent circuits. J. Appl. Phys. 1984, 55, 3148. [CrossRef]
Kim, S.J.; Cho, S.W.; Lee, H.; Lee, J.; Seong, T.Y.; Kim, I.; Park, J.-K.; Kwak, J.Y.; Kim, J.; Park, J.; et al. Frequency-tunable
nano-oscillator based on Ovonic Threshold Switch (OTS). arXiv:2009.13703.
Demirkol, A.S.; Ascoli, A.; Messaris, I.; Tetzlaff, R. Pattern formation dynamics in an MCNN structure with a numerically stable
VO2 memristor model. Jpn. J. Appl. Phys. under review.